Symmetries and conservation laws
Acta Wexionensia
No 170/2009
Mathematics
Symmetries and conservation laws
Raisa Khamitova
Växjö University Press
Symmetries and conservation laws. Thesis for the degree of Doctor of Philosophy, Växjö University, Sweden 2009.
Series editor: Kerstin Brodén
ISSN: 1404-4307
ISBN: 978-91-7636-650-9
Printed by: Intellecta Infolog, Göteborg 2009
Abstract
Conservation laws play an important role in science. The aim
of this thesis is to provide an overview and develop new methods
for constructing conservation laws using Lie group theory. The
derivation of conservation laws for invariant variational problems
is based on Noether’s theorem. It is shown that the use of LieBäcklund transformation groups allows one to reduce the number
of basic conserved quantities for differential equations obtained by
Noether’s theorem and construct a basis of conservation laws. Several examples on constructing a basis for some well-known equations are provided.
Moreover, this approach allows one to obtain new conservation
laws even for equations without Lagrangians. A formal Lagrangian
can be introduced and used for computing nonlocal conservation
laws. For self-adjoint or quasi-self-adjoint equations nonlocal conservation laws can be transformed into local conservation laws.
One of the fields of applications of this approach is electromagnetic theory, namely, nonlocal conservation laws are obtained for
the generalized Maxwell-Dirac equations. The theory is also applied to the nonlinear magma equation and its nonlocal conservation laws are computed.
Keywords: conservation law, Noether’s theorem, Lie group analysis, Lie-Bäcklund transformations, basis of conservation laws, formal Lagrangian, self-adjoint equation, quasi-self-adjoint equation,
nonlocal conservation law
v
Acknowledgments
My most sincere thanks are due to my supervisors Professor
Börje Nilsson and Assistant Professor Claes Jougréus for their
guidance and encouragement during my work on this thesis. My
thanks also go to Professor Andrej Khrennikov for offering valuable
advice during my studies as a PhD student.
I would like to express my greatest gratitude to Eva Pettersson,
head of the Department of Mathematics and Science, and Jan-Olof
Gustavsson, head of School of Engineering at Blekinge Institute of
Technology, for their support of my study.
I want to express my appreciation to Dr. Robert Nyqvist for his
help with LATEX. My thanks to my colleagues at the Department
of Mathematics and Science who showed interest towards my work
during this period. I would like to deeply thank the various people
who provided me with useful and helpful assistance.
Finally, my special thanks to my husband and our daughters
for all the good advice and their help and to my friend Nadezhda
Balayan for supporting me all these years.
vi
Contents
Abstract
Acknowledgments
Preface
1 Introduction
2 Conservation laws
2.1 Concept of a conservation law . . . . . . . . . . . . . .
2.2 Hamilton’s principle and the Euler-Lagrange equations
2.3 Lie group transformations and Noether’s theorem . . .
3 A basis of conservation laws
3.1 Classical mechanics . . . . . . . . . . . . . . . . . . . .
3.2 Relativistic mechanics . . . . . . . . . . . . . . . . . .
3.3 Motion in the de Sitter space . . . . . . . . . . . . . .
3.4 Nonlinear wave equation . . . . . . . . . . . . . . . . .
3.5 Lin–Reissner–Tsien equation . . . . . . . . . . . . . . .
3.6 Transonic three-dimensional gas motion . . . . . . . .
3.7 Short waves . . . . . . . . . . . . . . . . . . . . . . . .
3.8 Dirac equations . . . . . . . . . . . . . . . . . . . . . .
4 Equations without Lagrangians
4.1 Formal Lagrangian . . . . . . . . . . . . . . . . . . . .
4.2 Maxwell-Dirac equations . . . . . . . . . . . . . . . . .
4.3 Conservation laws . . . . . . . . . . . . . . . . . . . .
4.4 General magma equation . . . . . . . . . . . . . . . .
5 Summary of thesis
6 Summary of papers
6.1 Paper 1 . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Paper 2 . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Paper 3 . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Paper 4 . . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliography
Paper 1
Paper 2
Paper 3
Paper 4
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Preface
This thesis consists of two parts. The first part is an introductory section
which gives the reader some necessary background on the subject matter.
The thesis is based upon 4 papers included in the second part.
Papers included in the thesis:
1. On a basis of conservation laws of mechanics equations
R. Khamitova
Dokl. Akad. Nauk SSSR, 248(4), 1979, pp. 798–802.
English trans. Soviet Math. Dokl., 248(5), pp. 1071–1075, 1980.
2. Group structure and a basis of conservation laws
R. Khamitova
Teor. i Matem. Fizika, 52(2), pp. 244–251, 1982.
English trans. Theor. Math. Phys., 52(2), pp. 777–781, 1983.
3. Conservation laws for Maxwell-Dirac equations with dual Ohm’s law
N. Ibragimov, R. Khamitova and B. Thidé
J. Math. Phys., 48(5), pp. 053523-1–053523-11, 2007.
4. Symmetries and nonlocal conservation laws of the general magma equation
R. Khamitova
Com. Nonl. Sci. Num. Sim., 2008, doi: 10.1016/j.cnsns.2008.08.009
Contribution to the paper 3
In the paper 3 mentioned above, the author of this thesis calculated conservation laws.
Related papers not included in the thesis:
• On a basis of conservation laws in mechanics
R. Khamitova
Continuum Dynamics, 38, Novosibirsk, pp. 151–159, 1979 (In Russian).
• Adjoint system and conservation laws for symmetrized electromagnetic
equations with a dual Ohm’s law
N. Ibragimov, R. Khamitova and B. Thidé
Archives of ALGA, 3, ALGA publ., BTH, Karlskrona, Sweden,
pp. 81–95, 2006.
• Utilization of photon orbital angular momentum in the low-frequency
radio domain
B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi,
Ya. N. Istomin, N. H. Ibragimov, and R. Khamitova
Phys. Rev. Lett., 99, pp. 087701-1–087701-4, 2007.
• Self-adjointness and quasi-self-adjointness of the magma equation
R. Khamitova
1
in: J. A. Tenreiro Machado, M. F. Silva, R. S. Barbosa, L. B. Figueiredo
(Eds.), Proc. of the 2nd conference on "Nonlinear Science and Complexity", July 28–31, 2008, Porto, Portugal, ISBN:978-972-8688-56-1.
• Self-adjointness and quasi-self-adjointness of an equation modelling melt
migration through the Earth’s mantle. Nonlocal conservation laws
R. Khamitova
Preprint in Archives of ALGA, 5, ALGA publ., BTH, Karlskrona, Sweden, pp. 143–159, 2008.
2
1 Introduction
1 Introduction
Conservation laws and symmetries have always been of considerable interest
in science. They are important in the formulation and investigation of many
mathematical models. They were used, e.g. for proving global existence
theorems [1]–[3], in problems of stability [4], [5], in elasticity for studying
cracks and dislocations [6], [7], in astrophysics [8]–[10], in designing new
radio antennas [11] and so on (see also [12]).
Let us look at the use of symmetries and conservation laws, e.g. in celestial mechanics. In 1609, J. Kepler formulated two important laws known as
Kepler’s first and second laws. His first law states that the orbit of a planet
is an ellipse with the Sun as its focus. The second law says that if we join
the Sun and a planet by a straight line, the line will sweep out equal areas
at equal times.
What was important in these discoveries is that Kepler explained how the
planets moved. The next step, the explanation why they moved in such a
way, was given by I. Newton [13] in 1687. He formulated his law of gravity:
m1 m2
,
(1.1)
F =G
r2
where F is the force of gravity between two particles, G is a gravitational
constant, m1 and m2 are the masses of the particles and r is the distance
between them. At the beginning Newton tried to use a formula with r3
instead of r2 . However, he found out that it was not fruitful. When Newton
used the force of gravity (1.1) in his second law of motion, he obtained that
planets moved in ellipses. It proved to him that he was on the right track.
Thus, his way of discovery was by trial and error.
P.–S. Laplace showed that planets’ movement along ellipses followed from
the conservation law calculated by him, i.e. the conservation law for the
vector (see [14], Vol.1, Book II, Chap. III, Section 18):
x
(1.2)
A=v×M+μ ,
r
where v is the velocity of a planet, M = m(x×v) is the angular momentum,
m is the planet’s mass, x is a position-vector of the planet and r is the
magnitude of x. Laplace used the formal definition of a conservation law for
calculation of this conserved vector.
In 1983, N. H. Ibragimov [15] showed that it was possible to calculate
the vector (1.2) by using a certain symmetry of the Newton gravitational
field, a Lie-Bäcklund symmetry. This symmetry is more complicated than,
e.g. rotations, it depends not only on the position vector x but also on the
velocity v. Thus, the idea of symmetry and the corresponding conservation
law helps to explain the movement of planets in ellipses.
Kepler’s second law, the conservation of areas, follows from the conservation of angular momentum. This was established [16] independently by
L. Euler and D. Bernoulli. The angular momentum corresponds to the
cental symmetry of Newton’s gravitational field.
3
2 Conservation laws
There are several ideas for constructing conservation laws. One of them is
to use the direct method, when a conservation law for a differential equation
is derived by using its definition. As mentioned earlier, Laplace was the first
one who used this idea in 1798.
Another idea, that certain conservation laws for differential equations
obtained from a variational principle could appear from their symmetries,
followed from the works of Jacobi, Klein and Noether. In 1884, Jacobi [17]
showed a connection between conserved quantities and symmetries of the
equations of a particle’s motion in classical mechanics. Similar result was
obtained by Klein [18] for the equations of the general relativity. Klein predicted that a connection between conservation laws and symmetries could be
found for any differential equation obtained from a variational principle. He
suggested to Emmy Noether to investigate the possibility. She showed [19]
in 1918 that the conservation laws were associated with invariance of variational integrals with respect to continuous transformation groups. Noether
obtained the sufficient condition for existence of conservation laws. However, there are no explicit expressions for resulting conservation laws in
Noether’s work. In 1921, following Noether’s oral remark, Bessel-Hagen
[20] applied Noether’s theorem with the so-called "divergence" condition to
the Maxwell equations and calculated their conservations laws.
In 1951, Hill wrote a remarkable review paper [21] where he discussed
Noether’s theorem and presented the explicit formula for conservation laws
in the case of a first-order Lagrangian. The formula is written in terms of
variations (see [21], Eq. (43)). In 1969, inspired by Hill’s article, Ibragimov
[22] proved the generalized version of Noether’s theorem. In this theorem
conservations laws are related to the invariance of the extremal values of
variational integrals. He derived the necessary and sufficient condition for
existence of conservation laws. He also presented the explicit expressions
for calculating conservation laws in the case of a Lagrangian of any order.
On the basis of these theorems many conservations laws for differential
equations having a Lagrangian were calculated (see collected examples in
[23]–[25]).
2.1 Concept of a conservation law
Let us consider an ordinary differential equation
F (t, q, q̇, q̈) = 0
(2.1)
describing a motion of a dynamical system. Here t is time, q = (q 1 , ..., q s )
are the position coordinates, q = q(t), and v = q̇ ≡ dq
dt is the velocity,
q̈ =
4
d2 q
dt2 .
2 Conservation laws
Definition 2.1. A function C = C(t, q, v) is called a conserved quantity
for Eq. (2.1) if
dC
=0
(2.2)
dt
on every solution of Eq. (2.1).
In other words, the conserved quantity C(t, q, v) is constant on each trajectory q = q(t) and therefore is called a constant of motion.
In classical mechanics Eq. (2.1) has the form
mẍ = 0
(2.3)
and describes a free motion of a particle with the mass m and a position
vector x = (x1 , x2 , x3 ). The equation has several conserved quantities, e.g.
the energy E = 12 mv2 and the linear momentum p = mv.
Let us now consider a partial differential equation of p-th order
F (x, u, u(1) , u(2) , . . . , u(p) ) = 0
(2.4)
where the function F depends on n independent variables x, x = (x1 , . . . , xn ),
m dependent variables u, u = (u1 , . . . , um ), and the first, second, ..., p-th
order derivatives of u with respect to x denoted as u(1) = {uα
i }, u(2) =
α
{uα
ij }, . . . , u(p) = {ui1 i2 ...ip } respectively, α = 1, . . . , m and other indices
change from 1 to n.
Definition 2.2. A vector C = (C 1 , C 2 , . . . , C n ) where
C i = C i (x, u, u(1) , . . .),
i = 1, . . . , n,
is called a conserved vector for Eq. (2.4) if
div C = 0
(2.5)
on every solution of Eq. (2.4). We can also say that Eq. (2.5) is a conservation law for Eq. (2.4).
A conservation law for a system of partial differential equations can be
defined similarly.
Instead of dealing with functions uα = uα (x) and their derivatives, which
are also functions of x, one can treat all variables, x, u and derivatives of u, as
independent variables, called differential variables. Variables with the same
set of subscripts will be symmetric, for example uij = uji and so on. Using
the idea of differential variables [26] one can reformulate the definition of
a conservation law by introducing the operator of total differentiation with
respect to xi :
Di =
∂
∂
∂
∂
+ uα
+ uα
+ · · · + uα
+ ···
i
ij
ij1 ...jk
α
∂xi
∂uα
∂uα
∂u
j
j1 ...jk
(2.6)
where the usual convention of summation over repeated upper and lower
indices is used. Hence
(2.7)
div C (2.4) ≡ Di (C i )(2.4) = 0
5
where the notation (2.4) means that the relation holds on any solution
of Eq. (2.4). If one of the variables, for example x1 , is time t then the
component C 1 is called the density of the conservation law.
Remark 2.1. In practical calculations the conservation law (2.7) can be
rewritten to an equivalent form. If
1 + D2 (h2 ) + · · · + Dn (hn ).
C 1 (2.4) = C
then one obtains the following conservation law:
1 ) + D2 (C
2 ) + . . . + Dn (C
n ) = 0
Dt (C
where
2 = C 2 + Dt (h2 ),
C
...,
n = C n + Dt (hn )
C
because Dt Di (hi ) = Di Dt (hi ).
I have used this in my calculations of conservation laws.
By employing differential variables one can also rewrite Eq. (2.2) in the
following form:
dC ≡ Dt (C)(2.1) = 0.
(2.8)
dt (2.1)
Thus, conserved quantities and conserved vectors can be computed with
the help of Eq. (2.8) and Eq. (2.7), respectively (see, e.g. [27]–[31]).
2.2 Hamilton’s principle and the Euler-Lagrange equations
Consider again a motion of a dynamical system with a kinetic energy T (t, q, q̇)
and a potential energy U (t, q). The function
L(t, q, v) = T (t, q, q̇) − U (t, q)
is called the Lagrangian of the system.
Hamilton’s principle, or the principle of least action, states that the true
motion of the system between two chosen times t1 and t2 is described by
the fact that the trajectories of the particles provide an extremum of the
action functional
t2
L(t, q, v) dt.
(2.9)
t1
This requirement is equivalent to the statement that the Euler-Lagrange
equations:
∂L ∂L
−
D
= 0, α = 1, . . . , s
(2.10)
t
∂q α
∂v α
hold. They give a necessary condition for g(t) to provide an extremum of
the integral (2.9).
6
2 Conservation laws
In the case of several independent variables x = (x1 , . . . , xn ) and dependent variables u = (u1 , . . . , um ) an action integral has the form
L(x, u, u(1) , . . . , u(p) ) dx
(2.11)
V
where V is an arbitrary n-dimensional volume in the space of the variables
x and the Lagrangian L is a function depending on a finite number of
differential variables. The corresponding Euler-Lagrange equations have
the form:
δL
= 0, α = 1, ..., m,
(2.12)
δuα
where
∂
∂
δ
∂
=
− Di α + . . . + (−1)s Di1 Di2 . . . Dis α
+ . . . (2.13)
δuα
∂uα
∂ui
∂ui1 i2 ...is
is the variational derivative.
In my first two articles I discuss conservation laws for the Euler-Lagrange
equations.
Definition 2.3. A conservation law is called a trivial conservation law if
Di (C i ) ≡ 0
δL
δL
or C i are smooth functions of δu
α , Di δuα , . . . . Two conservation laws which
only differ by a trivial conservation law are regarded as equivalent.
2.3 Lie group transformations and Noether’s theorem
Assume that the Euler-Lagrange equations (2.12) admit a one-parameter
Lie transformation group G, i.e. a local group of transformations
x̄ = ϕ(x, u, a),
where
ϕ = (ϕ1 , ..., ϕn ),
ū = ψ(x, u, a),
ψ = (ψ 1 , ..., ψ m ),
and
ϕ(x, u, 0) = x,
ψ(x, u, a) = u.
The infinitesimal generator of the group G has the form
X = ξ i (x, u)
∂
∂
+ η α (x, u) α ,
∂xi
∂u
(2.14)
where
ξ i (x, u) =
∂ϕi (x, u, a) ,
∂a
a=0
η α (x, u) =
∂ψ α (x, u, a) .
∂a
a=0
7
Definition 2.4. A variational integral (2.10) is invariant under the group
G if
L(x̄, ū, ū(1) , . . . , ū(p) ) dx̄ =
L(x, u, u(1) , . . . , u(p) ) dx.
V
V
The invariance condition is given by the following lemma.
Lemma 2.1. An integral (2.11) is invariant under the group G if and only
if [15]
X(L) + LDi (ξ i ) = 0.
(2.15)
Here X is a prolonged version of the generator (2.14):
X = ξi
∂
∂
∂
∂
+ η α α + ζiα α + · · · + ζiα1 ...is α
+ ··· ,
∂xi
∂u
∂ui
∂ui1 ...is
(2.16)
where
j α
ζiα = Di (η α − ξ j uα
j ) + ξ uji ,
j α
ζiα1 ...is = Di1 ...Dis (η α − ξ j uα
j ) + ξ uji1 ...is .
Noether proved her theorem by the application of the variational procedure to the integral of action. Using her idea Hill presented the explicit form
of conserved quantities in the case of the first-order Lagrangians L(x, u, u(1) )
(see [21], Eq. (43)). In my articles I have used the following generalized
form of Noether’s theorem proved by Ibragimov [22], [32] on the basis of
the group-theoretical approach.
Theorem 2.1. Let the variational integral (2.11) be invariant with respect
to a group G with generators (2.14). Then a vector C with components
C i = N i (L),
i = 1, 2, ..., n,
(2.17)
is a conserved vector for the Euler-Lagrange equations (2.12), i.e.
= 0.
Di (C i )
2.12)
Here N i are Ibragimov’s operators [32], [15]:
∂
∂
N i = ξi + W α
+
(−1)s Dj1 ...Djs α
α
∂ui
∂uij1 ...js
+
r≥1
Dk1 ...Dkr (W α )
(2.18)
(2.19)
s≥1
∂
∂uα
ik1 ...kr
+
(−1)s Dj1 ...Djs
s≥1
∂
∂uα
ik1 ...kr j1 ...js
,
where W α = η α − ξ j uα
j.
Corollary. If for some one-parameter transformation group the invariance
condition (2.15) is not satisfied but the "divergence" condition
X(L) + LDi (ξ i ) = Di (B i )
(2.20)
holds, then the components of the corresponding conserved vector have the
form:
(2.21)
C i = N i (L) − B i , i = 1, 2, ..., n.
8
3 A basis of conservation laws
3 A basis of conservation laws
Besides the operator X, we shall also use its equivalent canonical LieBäcklund operator [33], [34]:
X = X − ξ j Dj = η̄ α
∂
∂
∂
+ ζ̄iα α + · · · + ζ̄iα1 ...is α
+ ··· ,
∂uα
∂ui
∂ui1 ...is
(3.1)
where
η̄ α = W α = η α − ξ j uα
j,
ζ̄iα = Di (η̄ α ) ,
ζ̄iα1 ...is = Di1 ...Dis (η̄ α ).
Some conservation laws can be obtained more readily by using the commutativity of the generators X (or generators (2.14) with ξ 1 = const., ...,
ξ n = const. ) and Di .
Lemma 3.1. A canonical Lie–Bäcklund operator X and an operator of
total differentiation Di are commutative [15]:
XDi = Di X.
Lemma 3.2. If C = (C 1 , ..., C n ) satisfies a conservation law for some differential equation and a generator X is admitted by the equation in question
then the vector with the components
C i = X(C i )
(3.2)
also satisfies a conservation law [15].
Hence lemmas 3.1 and 3.2 furnish the basis for another idea for calculating conservation laws. Moreover, conserved vectors can be computed for a
differential equation without any Lagrangian if it has a known conservation
law (see, e.g. [35] and [36]).
The property (3.2) makes it possible to introduce the concept of a basis
(with respect to the group G) of the conservation laws and thus reduce
the number of vectors C that must be constructed by means of Noether’s
theorem.
Definition 3.1. Let {C} be a set of vectors satisfying the conservation law
(2.18). A basis of the set {C} is its minimal subset from which {C} can be
obtained by repeated application of (3.2) and by linear combinations. The
conservation laws corresponding to the basis vectors form the basis of the
conservation laws.
Using an example of gasdynamics equations, it was conjectured in [32]
that the following diagram is commutative:
X
⏐1
i⏐
N1 C1
adX
−→ X
⏐2
⏐ i
N2
X
−→
C2
9
and this statement can be used for construction of a basis of conserved
vectors. The operators N1i and N2i in the diagram are given by (2.19) and
X, X1 , X2 by (2.14), the action adX is defined as follows:
adX(X1 ) ≡ [X, X1 ] = XX1 − X1 X.
Hence the commutator [X, X1 ] has the form
∂
∂
+ X(η1α ) − X1 (η α )
·
[X, X1 ] = X(ξ1i ) − X1 (ξ i )
i
∂x
∂uα
(3.3)
Following this idea I verified the validity of the statement by means of
several examples [37], [38] and then proved the following general result [39].
Theorem 3.1. Let generators X, X1 , X2 of the form (2.14) be admitted
by the Euler-Lagrange equations (2.12). Let the conserved vectors C1 , C2
correspond (by Noether’s theorem) to the generators X1 , X2 and let
[X, X1 ] = X2 .
Then the vectors X(C1 ) and C2 define equivalent conserved vectors, i.e.
X(C1 ) = C2 .
Remark 3.1. The theorem also holds when instead of the invariance condition (2.15) of a variational integral we have the "divergence" condition
(2.20.)
The proof of the theorem is given in [39]. Later this theorem was formulated in another form in [12]. Specific examples given in [37] were used by
Tsujishita [40] as applications in modern formal differential geometry.
Let us consider several examples.
3.1 Classical mechanics
Let Eq. (2.3),
mẍ = 0, x = (x1 , x2 , x3 ),
describe the free motion of a particle of mass m. The equation has the
Lagrangian
1
L = m|ẋ|2
2
and admits a 10-parameter point transformation group containing space
translations with the generators
Xμ =
∂
,
∂xμ
μ = 1, 2, 3,
(3.4)
∂
,
∂t
(3.5)
translation of time with the generator
X4 =
10
3 A basis of conservation laws
rotations of the vector x with the generators
Xμν = xν
∂
∂
− xμ ν ,
μ
∂x
∂x
μ, ν = 1, 2, 3,
(3.6)
and Galilean transformations with the generators
Xμ4 = t
∂
,
∂xμ
(μ = 1, 2, 3).
(3.7)
Hence according to Noether’s theorem Eq. (2.3) has 10 conservation laws
of the form
Dt (C)(2.3) = 0
(3.8)
defined by the following conserved quantities: the linear momentum
p = mẋ,
the energy
E=
1
m|ẋ|2
2
the angular momentum
M=p×x
and the vector
q = m(x − ẋt).
From the table of commutators, Table 3.1, it is easy to notice that only
X4 and one of generators Xμν can not be obtained by using adX. Thus,
employing Theorem 3.1 we can conclude the following:
A basis of conservation laws consists of two conservation laws defined by
the energy E and one of the components of the angular momentum M.
Indeed, if we choose as a basis of conserved quantities E and, e.g. M 1 we
can obtain other conserved quantities by means of the generators Xμ4 and
Xμν written in the prolonged form:
Xμ4 = t
∂
∂
+
∂xμ
∂ ẋμ
(μ = 1, 2, 3)
and
Xμν = xν
∂
∂
∂
∂
− xμ ν + ẋν μ − ẋμ ν
μ
∂x
∂x
∂ ẋ
∂ ẋ
(μ, ν = 1, 2, 3).
(3.9)
Then
Xμ4 E = pμ ,
X12 (M 1 ) = M 2 ,
X34 M 2 = q 1 ,
X24 M 1 = q 3 ,
X13 (M 1 ) = M 3 .
X14 M 3 = q 2 .
11
Table 3.1: Table of commutators (Classical mechanics)
X1
X1
X2
X3
X4
X12
X23
X13
X14
X24
X34
0
0
0
0
−X2
0
−X3
0
0
0
0
0
0
X1
−X3
0
0
0
0
0
0
0
X2
X1
0
0
0
0
0
0
0
X1
X2
X3
0
−X13
X23
X24
−X14
0
0
−X12
0
X34
−X24
0
X34
0
−X14
0
0
0
0
0
X2
X3
X4
X12
X23
X13
X14
X24
0
X34
3.2 Relativistic mechanics
The equation of free motion of a relativistic particle in the Minkowski space
with the metric
ds2 = c2 dt2 − dx2 − dy 2 − dz 2
has the Lagrangian
L = −mc
2
1−
|v|2
,
c2
vi =
dxi
dt
(i = 1, 2, 3),
where c is a constant equal to the light velocity in vacuum,
x1 = x, x2 = y, x3 = z.
It admits the 10-parameter non-homogeneous Lorentz group with the generators (3.4)–(3.6) and the generators of the Lorentz transformations
Xμ4 = x4
∂
∂
1
+ 2 xμ 4 ,
∂xμ
c
∂x
(μ = 1, 2, 3)
(3.10)
where x4 = t.
The corresponding conservations laws have the form similar to (3.8). According to Noether’s theorem they are defined by the following conserved
quantities:
p0 = mc ẋ,
12
E0 = mc3 ẋ4 ,
M0 = p0 × x,
Q0 = mc(xẋ4 − ẋx4 ) (3.11)
3 A basis of conservation laws
Table 3.2: Table of commutators (Relativistic mechanics)
X1
X1
X2
X3
X4
X12
X23
X13
X14
X24
X34
0
0
0
0
−X2
0
−X3
1
c2 X4
0
0
0
0
0
X1
−X3
0
0
1
c2 X4
0
0
0
0
X2
X1
0
0
1
c2 X4
0
0
0
0
X1
X2
X3
0
−X13
X23
X24
−X14
0
0
−X12
0
X34
−X24
0
X34
0
−X14
0
−1
c2 X12
−1
c2 X13
0
−1
c2 X23
X2
X3
X4
X12
X23
X13
X14
X24
0
X34
where x = (x1 , x2 , x3 ) and the dot denotes differentiation with respect to
the length of the arc s in the Minkowski space. Comparing Table 3.2 and
Table 3.1, one can see a significant difference. Namely, in the case of relativistic mechanics the time translation generator X4 can be obtained from
other operators by using adX. Therefore, employing Theorem 3.1 we can
conclude that
A basis of conserved quantities (3.11) (with respect to group G) is defined
by one conserved quantity, e.g. any of the components of the angular momentum M0 .
Indeed, if we choose M01 as a basis of conserved quantities, under the
action of the generators of the Lorentz transformations written in the prolonged form
Xμ4 = x4
∂
∂
∂
∂
1
1
+ 2 xμ 4 + x˙4 μ + 2 ẋμ 4 ,
∂xμ
c
∂x
∂ ẋ
c
∂ ẋ
(μ = 1, 2, 3)
(3.12)
and the generators of rotation (3.9) we obtain:
X24 (M01 ) = Q30 ,
Then we have
X12 (M01 ) = M02 ,
X34 (M02 ) = Q10 ,
X13 (M01 ) = M03 .
X14 (M03 ) = Q20 ,
E0 can be obtained from Q0 with the help of the translation generators Xμ ,
13
i.e.
c2 Xμ (Qμ0 ) = E0
and the energy E0 transforms into the momentum p0 ,
μ
Xμ4 (E0 ) = p0 .
Remark 3.2. On the other hand it is also possible to choose any component
of the vector Q0 as a basis of conserved quantities.
3.3 Motion in the de Sitter space
Consider the space V4 with the metric
ds2 =
1 2 2
(c dt − dx2 − dy 2 − dz 2 ),
Φ2
(3.13)
where
K 2
r , r2 = c2 t2 − x2 − y 2 − z 2
(3.14)
4
and K = const. denotes the curvature of the Sitter space-time.
As well as the equation of free motion of a particle in Minkowski space a
similar equation in the de Sitter space has the Lagrangian
|v|2
dxi
mc2
(μ = 1, 2, 3),
1 − 2 , vμ =
L=−
Φ
c
dt
Φ=1+
also admits 10-parameter group G with the generators of rotations and the
generators of Lorentz transformations of the form
Xμν = xν
∂
∂
− xμ ν
∂xμ
∂x
(μ < ν,
μ = 1, 2, 3; ν = 1, 2, 3, 4),
(3.15)
but the generators of space translations (3.4) and translations of time (3.5)
are replaced by the generators
K
∂
xν xi + (Φ − 2)δ νi
, ν, i = 1, 2, 3, 4.
(3.16)
Xν =
2
∂xi
Here the generators are written in the coordinates
x1 = x,
x2 = y,
x3 = z,
x4 = ict
and δ ki is a Kronecker symbol.
According to Noether’s theorem there are 10 conserved quantities similar
to (3.11), the linear momentum pk ,, the energy EK , the angular momentum
MK and the vector QK .
The structure of Lie algebra with the basis (3.15)–(3.16) is determined
by the commutators
[Xμ , Xν ] = KXμν ,
[Xμ , Xμν ] = Xν ,
[Xμν , Xα ] = 0 (α = μ, α = ν),
[Xμν , Xαβ ] = δμα Xνβ + δνβ Xμα − δμβ Xνα − δνα Xμβ .
14
3 A basis of conservation laws
Hence, for the equation of free motion of a particle in the de Sitter space,
we arrive at the following assertion:
A basis of conserved quantities with respect to the group G is defined by one
conserved quantity, e.g. any of the components of the angular momentum
MK = M0 /Φ2 .
Remark 3.3. In this case any conserved quantity, i.e the energy, or any
component of the linear momentum or any component of the vector QK
can also be chosen as a basis of the conserved quantities.
3.4 Nonlinear wave equation
The equation
utt − Δu + λu3 = 0
(3.17)
has the Lagrangian
1
L = |Δu|2 − u2t + λu4
2
where Δu = uxx + uyy + uzz , λ = const.. Eq. (3.17) describes string vibration immersed in nonlinear medium. Eq. (3.17) is also used in quantum
nonlinear field theory. It admits the 15-dimensional group of conformal
transformations in the Minkowski space and has, correspondingly, 15 conservation laws of the form
Dt (C 1 ) + Dx (C 2 ) + Dy (C 3 ) + Dz (C 4 ) = 0.
The basis of conserved vectors for the nonlinear wave equation also consists
of one conserved vector [37], [38].
3.5 Lin–Reissner–Tsien equation
The equation [41]
−ϕx ϕxx − 2ϕxt + ϕyy = 0
describes the non-steady-state potential gas flow with transonic velocities.
It has the Lagrangian
1
L = |Δu|2 − u2t + λu4 ,
2
Δu = uxx + uyy ,
λ = const.
and admits an infinite transformation group [42]. Accordingly, by Noether’s
theorem, the family of conservation laws [43] is infinite. Meanwhile, employing Theorem 3.1 we can conclude the following:
The basis of conservation laws consists of one conservation law [37], namely
Dt (C 1 ) + Dx (C 2 ) + Dy (C 3 ) = 0
where
C1 =
1 2 1 3
ϕ − ϕ ,
2 y 6 x
1
C 2 = ϕ2t + ϕ2x ϕt ,
2
C 3 = −ϕt ϕy .
15
3.6 Transonic three-dimensional gas motion
The equation
−ux uxx − 2uxt + uyy + uzz = 0
of transonic gas motion has the Lagrangian
L=
1 3
1
1
u + ux ut − u2y − u2z
6 x
2
2
and admits an infinite group transformations (the generators [44], [45], [39]
depends on arbitrary functions).
The basis of conservation laws
Dt (C 1 ) + Dx (C 2 ) + Dy (C 3 ) + Dz (C 4 ) = 0
is defined by two vectors A1 and A4 where
1
1
1
1
u2x +ut , A31 = ut uy , A41 = ut uz ;
A11 = u3x − u2y − u2z , A21 = −ut
6
2
2
2
1
u2 + ut , A34 = −η̄uy + zL, A44 = −η̄uz − yL,
A14 = η̄ux , A24 = η̄
2 x
η̄ = yuz − zuy .
3.7 Short waves
During first underwater nuclear and thermonuclear explosions near the arctic island Novaja Zemlja in the USSR it was discovered that weak waves
were drastically increasing the destructive force of a shock wave [46]. Rizhov
and Khristianovich [47] presented the equations describing the behavior of
these so-called "short waves".
The equations of short waves
uy − 2vt − 2(v − x)vx − 2kv = 0,
vy + ux = 0,
k = const.,
admit an infinite-dimensional group [48]. They can be reduced by the substitution u = ϕy , v = −ϕx to the equation
ϕyy + 2ϕxt − 2(x + ϕx )ϕxx + 2kϕx = 0,
(3.18)
which has the Lagrangian
1
1
L = (ϕt ϕx − ϕ3x − xϕ2x + ϕ2y ) exp[2(k + 1)t].
3
2
In [39] I calculated the following generators for Eq. (3.18):
∂
∂
∂
∂
− 4(k + 1)x
− 2(k + 1)y
− 8(k + 1)ϕ ,
∂t
∂x
∂y
∂ϕ
∂
∂
∂
y3
X2 = −μ y
+μ
− xy(μ + μ ) −
dk μ
,
∂x
∂y
3
∂ϕ
X1 = 9
16
(3.19)
(3.20)
3 A basis of conservation laws
∂
∂
+ y 2 dk κ − x(κ + κ)
,
(3.21)
∂x
∂ϕ
∂
∂
∂
X4 = λy
, X5 = σ
, X0 = y
(3.22)
∂ϕ
∂ϕ
∂t
where μ, κ, λ, σ are arbitrary functions of t, the prime denotes differentiation
with respect to t, dk = d2 /dt2 + (k + 1)d/dt + k.
Although the constant k in Eq. (3.18) takes only the values 0 and 1 in
accordance with the physical content of the problem, it can be regarded as
an arbitrary parameter. For k = 2; 1/2 there is an extension of the group,
the following generators are added:
X3 = κ
X6 = 9a
∂
∂
∂
+{[3a −4(k+1)a]x−[3a −(k+1)a ]y 2 } +[6a −2(k+1)a]y
∂t
∂x
∂y
x2 + −[3a +8(k+1)a]ϕ− [3a +(5−4k)a ]+[3a +(2−k)a −(k+1)a ]xy 2
2
∂
y4
·
− [3aiv + 2(k + 1)a − (k 2 − k + 1)a − k(k + 1)a ]
6
∂ϕ
The operator X0 does not satisfy the conditions of Noether’s theorem and
the generators X4 and X5 give only trivial conservation laws.
Among the commutation relations for X1 , X2 , X3 , X6 we can distinguish
ad X2 (X1 ) = −X2 < 9μ + 2(k + 1)μ >,
ad X3 (X1 ) = −X3 < 9κ + 4(k + 1)κ >,
ad X6 (X1 ) = −X6 < 9a > .
Here the brackets < ... > mean that instead of the arbitrary function occurring in the coordinates of the generator (or conserved vector) it is necessary
to substitute the expression in these brackets.
Therefore, the basis of the conservation laws
Dt (C 1 ) + Dx (C 2 ) + Dy (C 3 ) = 0
is determined by one vector corresponding to X1 , i.e. A1 with coordinates
A11 = η̄1 Eϕx + 9L,
A31 = η̄1 Eϕy −2(k+1)yL,
A21 = η̄1 E(ϕt − ϕ2x − 2xϕx ) − 4(k + 1)xL,
η̄1 = −8(k+1)ϕ−9ϕt +4(k+1)xϕx +2(k+1)yϕy .
3.8 Dirac equations
The Dirac equations
∂ψ
∂ ψ k
+ mψ = 0,
γ − mψ = 0,
(3.23)
k
∂x
∂xk
are the relativistic quantum mechanical wave equations used for the description of fermions, i.e. elementary particles having half-integer spin number
(say, 12 , 32 , 52 , ...). Eqs (3.23) have the Lagrangian
γk
L=
∂ ψ
1 k ∂ψ
+ mψ −
γ k − mψ ψ ,
ψ γ
k
k
2
∂x
∂x
k = 1, 2, 3, 4.
17
Here the independent variables are
x1 = x, x2 = y, x3 = z, x4 = ict,
the dependent variables
ψ = (ψ 1 , ..., ψ 4 ) ψ = (ψ1 , ..., ψ4 )
are 4-dimensional complex vectors and and γ k are 4 × 4 complex matrices.
The maximal group admitted by Eqs. (3.23) is obtained in [22]. The
Dirac equations have an infinite number of conservation laws
Dt (C 1 ) + Dx (C 2 ) + Dy (C 3 ) + Dz (C 4 ) = 0.
Using Theorem 3.1 we obtain the following result [39].
For m = 0 the basis of conserved vectors is formed by 3 vectors, A12 , A5 , A8 ,
and for m = 0 by 2 vectors A12 , A5 .
Their coordinates have the form
Ak12 =
1 k 1 2
{ψ(γ γ γ + γ 1 γ 2 γ k )ψ} + x2 Ak1 − x1 Ak2 ,
4
k ψ,
Ak5 = −iψγ
k γ 5 ψ,
Ak8 = iψγ
where
Akl =
1 ∂ ψ k
k ∂ψ + δ k ,
γ ψ − ψγ
l
l
l
2 ∂x
∂x
k = 1, 2, 3, 4, l = 1, 2,
and δlk is a Kronecker’s symbol.
4 Equations without Lagrangians
4.1 Formal Lagrangian
Many differential equations cannot be formulated as the Euler–Lagrange
equations since they have no Lagrangians. Therefore, it is impossible to
apply Noether’s theorem for calculating conservation laws. However, according to [49] and [50], it is possible to introduce a formal Lagrangian if
any given system of equations is taken into consideration together with the
adjoint system. In his recent paper [50] Ibragimov has proved that the adjoint system inherits symmetries of the given system and has suggested a
new theorem on nonlocal conservation laws.
Consider an arbitrary system of sth-order partial differential equations
(4.1)
Fα x, u, u(1) , . . . , u(s) = 0, α = 1, . . . , m.
where the functions Fα (x, u, u(1) , . . . , u(s) ) depend on n independent variables x = (x1 , . . . , xn ), m dependent variables u = (u1 , . . . , um ), u = u(x),
and their derivatives up to an arbitrary order s.
18
4 Equations without Lagrangians
Definition 4.1. The adjoint system to Eqs (4.1) is defined by [51]
δ(v β Fβ )
= 0,
Fα∗ x, u, v, u(1) , v(1) , . . . , u(s) , v(s) ≡
δuα
α = 1, . . . , m, (4.2)
where v = (v 1 , . . . , v m ) are new dependent variables, v = v(x), and
the variational derivative (2.13).
δ
δuα
is
In the case of linear equations this definition is equivalent to the standard
one.
Remark 4.1. The variables v = (v 1 , . . . , v m ) were called in [50] nonlocal
variables in accordance with the general concept of nonlocal symmetries.
Therefore, conservation laws involving v were named nonlocal conservation
laws.
Using the new definition of the adjoint system, it can be shown that any
system of sth-order differential equations (4.1) considered together with
its adjoint equation (4.2) has a Lagrangian. Namely, the Euler-Lagrange
equations with the Lagrangian
(4.3)
L = v β Fβ x, u, u(1) , . . . , u(s)
provide the simultaneous system of equations (4.1), (4.2) with 2m dependent
variables u = (u1 , . . . , um ) and v = (v 1 , . . . , v m ).
Definition 4.2. The system (4.1) is called self-adjoint if the substitution
v = u gives
(4.4)
F ∗ = λ(x, u, u(1) , . . . , u(s) )F.
The system (4.1) is called quasi-self-adjoint [52] if there exists a function
h(u) such that Eq (4.4) holds upon the substitution v = h(u).
4.2 Maxwell-Dirac equations
We have the system of equations
∇×E+
∂B
+ σm B = 0,
∂t
∇×B−
∂E
− σe E = 0,
∂t
(4.5)
∇ · E − ρe = 0,
∇ · B − ρm = 0,
where σm , σe = const. The system (4.5) has eight equations for eight dependent variables: six coordinates of the electric and magnetic vector fields
E = (E 1 , E 2 , E 3 ) and B = (B 1 , B 2 , B 3 ), respectively, and two scalar quantities ρe and ρm , the electric and magnetic monopole charge densities.
19
Using (4.3) we write the Lagrangian (4.3) for Eqs. (4.5) in the following
form [54] :
∂B
+ σ m B + R e ∇ · E − ρe
L=V· ∇×E+
∂t
∂E
− σ e E + R m ∇ · B − ρm ,
(4.6)
+W · ∇ × B −
∂t
where V, W, Re , Rm are adjoint variables. With this Lagrangian the
adjoint equations for the new dependent variables V, W, Re , Rm have the
form [55]
∂W
− σe W = 0,
∇×V+
∂t
∇×W−
∂V
+ σm V = 0,
∂t
Re = 0,
(4.7)
Rm = 0.
4.3 Conservation laws
Each generator
X = ξ i (x, u)
∂
∂
+ η α (x, u) α ,
∂xi
∂u
admitted by a first-order system
Fα (x, u, u(1) ) = 0,
α = 1, ..., m
leads to a conserved vector with the components
∂Fβ )
C i = v β ξ i Fβ + (η α − ξ j uα
j
∂uα
i
(4.8)
where i = 1, ..., n and v β solve the adjoint system
Fα∗ (x, u, u(1) , v(1) ) = 0,
α = 1, ..., m.
The conservation law for Eqs (4.5) has the form
Dt (τ ) + div χ = 0,
(4.9)
which holds on the solutions of Eqs (4.5) and (4.7). Here τ is the density
of the conservation law (4.9), χ = (χ1 , χ2 , χ3 ), and
div χ ≡ ∇ · χ = Dx (χ1 ) + Dy (χ2 ) + Dz (χ3 ).
The Maxwell-Dirac equations are neither self-adjoint nor quasi-self-adjoint.
Consequently, the conservation laws obtained by using Eqs (4.8) are nonlocal.
20
5 Summary of thesis
4.4 General magma equation
The equation
∂ n
∂f
∂ 1 ∂f +
f 1−
=0
∂t
∂z
∂z f m ∂t
(4.10)
models the migration of melt through the Earth’s mantle. It follows from
the equations
∂ 1 ∂u uz = −ft , u = f n 1 +
(4.11)
∂z f m ∂z
where u is the vertical barometric flux of melt, f is the volume fraction
of melt, z is a vertical space coordinate and t is time. All the variables
are dimensionless. Eqs. (4.11) were proposed by Scott and Stevenson [56].
They suggested that 2 ≤ n ≤ 5 or even bigger and supposed that 0 ≤ m ≤ 1.
Some authors discussed Eq. (4.10) for any values of n and m.
I denote f by u in order to make Eq. (4.10) compatible with the general
notation used above. It has the form
(4.12)
F ≡ ut + Dz un 1 − Dz u−m ut . = 0
The general magma equation does not have any Lagrangian and therefore
the formal Lagrangian is introduced. Using the Lagrangian and employing
infinitesimal symmetries of Eq. (4.10) nonlocal conservation laws are obtained in my articles [57]– [59]. The central part of these articles is the
proof of the remarkable property of Eq. (4.12) to be quasi-self-adjoint for
any values of the parameters m and n. This property allows us to obtain
local conservation laws from nonlocal ones. They include the local conservation laws obtained by the direct method by Barcilon and Richter [27] and
Harris [28] and later discussed in [36].
5 Summary of thesis
This thesis is a collection of four papers. I present the outline of each of them
below. My investigations in this thesis concern symmetries and conservation
laws. Two ideas are discussed throughout the text. The first one is about
constructing a basis of conservation laws (with respect to admitted group
G) obtained from Noether’s theorem (papers 1 and 2). It is proven that
there is a connection between a basis of conservation laws and the structure
of the Lie algebra of G.
The second idea is the contraction of conservation laws for differential
equations without Lagrangians (papers 3 and 4). Nonlocal conservation
laws are constructed for Dirac’s symmetrized Maxwell-Lorentz equations
with dual Ohm’s law and the equation modelling a melt migration through
the Earth’s mantle.
21
6 Summary of papers
6.1 Paper 1
Lie-Bäcklund groups allow us to reduce a number of independent conserved
quantities for differential equations and construct a basis of conservation
laws. This article is devoted to constructing a basis for some equations in
mechanics by the method described in [32]. The results are the following:
for the equation of free motion of a particle in classical mechanics a basis
(with respect to the admitted group G) of the set of 10 conserved quantities
consists of two conserved quantities, namely the energy and one of the
components of the angular momentum.
For the motion of a relativistic particle a basis of the set of 10 conserved
quantities is formed by one conserved quantity. In the case of the Minkowski
space it can be any component of the angular momentum M0 or any component of the vector Q0 , in the case of the de Sitter space it can be any
of the conserved quantities, e.g. any component of the angular momentum
MK .
The nonlinear wave equation, describing string vibration immersed in
nonlinear media, has a set of 15 conserved vectors. The basis is defined by
one conserved vector.
The Lin-Reissner-Tsien equation for the non-steady-state potential gas
flow with transonic velocities possesses an infinite set of conservation laws.
The basis consists of one conservation law.
6.2 Paper 2
The derivation of conservation laws for invariant variational problems is
based on fundamental identity connecting Lie-Bäcklund, Euler-Lagrange
and Ibragimov’s operators. It is shown that this identity also makes it possible to establish a connection between a basis of conservation laws (with
respect to the group G admitted by the considered system of differential
equations) and the structure of the Lie algebra of G. This provides a justification for the basis construction scheme proposed by Ibragimov. The
theorem is applied to the short waves equation, to the transonic threedimensional gas motion equation and the Dirac equations. As a result a
basis of conservation laws for each equation is obtained.
6.3 Paper 3
Some differential equations have no Lagrangian. However using a general
theorem on conservation laws for arbitrary differential equations proven by
Ibragimov [50] it is possible to introduce a formal Lagrangian. We have derived conservation laws for Dirac’s symmetrized Maxwell-Lorentz equations
under the assumption that both the electric and magnetic charges obey linear conductivity laws (dual Ohm’s law). The conserved quantities obtained
in the article involve solutions v and w of the adjoint equations. It may be
useful for applications to use an alternative representation of the conserved
22
6 Summary of papers
quantities in terms of the electric and magnetic vector fields E and B only.
It is shown that nonlocal conservation laws (see Remark 4.1) can be written
in two-solution or one-solution representations.
6.4 Paper 4
A general magma equation models a melt migration through the Earth’s
mantle. It is a nonlinear third-order differential equation which does not
have any Lagrangian. Therefore, the recent theorem on nonlocal conservation laws [50] is applied to this equation. It is shown that the equation has a remarkable property: for any values of the parameters m and n
the general magma equation is quasi-self-adjoint in the terminology of [52].
The self-adjoint equations are also singled out. Nonlocal conservation laws
are obtained using the symmetries of the equation. By employing quasiself-adjointness of the equation it is shown that all local conservation laws
calculated by direct method are consequences of the nonlocal ones.
23
Bibliography
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28
Paper I
29
Paper II
37
Paper III
45
JOURNAL OF MATHEMATICAL PHYSICS 48, 053523 共2007兲
Conservation laws for the Maxwell-Dirac equations
with dual Ohm’s law
Nail H. Ibragimova兲 and Raisa Khamitova
Department of Mathematics and Science, Research Centre ALGA: Advances in Lie Group
Analysis, Blekinge Institute of Technology, SE-371 79 Karlskrona, Sweden
Bo Thidéb兲
Ångström Laboratory, Swedish Institute of Space Physics, P.O. Box 537, SE-751 21
Uppsala, Sweden
共Received 5 March 2007; accepted 9 April 2007; published online 31 May 2007兲
Using a general theorem on conservation laws for arbitrary differential equations
proved by Ibragimov 关J. Math. Anal. Appl. 333, 311–320 共2007兲兴, we have derived
conservation laws for Dirac’s symmetrized Maxwell-Lorentz equations under the
assumption that both the electric and magnetic charges obey linear conductivity
laws 共dual Ohm’s law兲. We find that this linear system allows for conservation laws
which are nonlocal in time. © 2007 American Institute of Physics.
关DOI: 10.1063/1.2735822兴
I. INTRODUCTION
In all areas of physics, conservation laws are essential since they allow us to draw conclusions
of a physical system under study in an efficient way.
Electrodynamics, in terms of the standard Maxwell electromagnetic equations for fields in
vacuum, exhibit a rich set of symmetries to which conserved quantities are associated. Recently,
there has been a renewed interest in the utilization of such quantities. Here we use a theorem of
Ibragimov 共2006兲 to derive conservation laws for Dirac’s symmetric version of the MaxwellLorentz microscopic equations, allowing for magnetic charges and magnetic currents, where the
latter, just as electric currents, are assumed to be described by a linear relationship between the
field and the current, i.e., Ohm’s law. The method of Ibragimov 共2006兲 produces two new adjoint
vector fields which fulfil, Maxwell-like equations. In particular, we obtain conservation laws for
the symmetrized electromagnetic field which are nonlocal in time.
II. PRELIMINARIES
A. Notation
We will use the following notation 共see, e.g., Ibragimov, 1999兲. Let x = 共x1 , . . . , xn兲 be independent variables and u = 共u1 , . . . , um兲 be dependent variables. The set of the first-order partial
derivatives u␣i = ⳵u␣ / ⳵xi will be denoted by u共1兲 = 兵u␣i 其, where ␣ = 1 , . . . , m and i , j , ¯ = 1 , . . . , n.
The symbol Di denotes the total differentiation with respect to the variable xi:
Di =
⳵
⳵
⳵
⳵
+ u␣i ␣ + u␣ij ␣ + u␣ijk ␣ + ¯ .
⳵u
⳵xi
⳵u j
⳵u jk
We employ the usual convention of summation in repeated indices.
Recall that a necessary condition for extrema of a variational integral
a兲
Electronic mail: [email protected]
Also at LOIS Space Centre, School of Mathematics and Systems Engineering, Växjö University, SE-351 95 Växjö,
Sweden; electronic mail: [email protected]
b兲
0022-2488/2007/48共5兲/053523/11/$23.00
48, 053523-1
© 2007 American Institute of Physics
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053523-2
J. Math. Phys. 48, 053523 共2007兲
Ibragimov, Khamitova, and Thidé
冕
L共x,u,u共1兲兲dx,
共1兲
V
with a Lagrangian L共x , u , u共1兲兲, depending on first-order derivatives, is given by the EulerLagrange equations
冉 冊
␦L
⳵L
⳵L
⬅
− Di
= 0,
␦u␣ ⳵u␣
⳵u␣i
␣ = 1, . . . ,m.
共2兲
We will understand by a symmetry of a certain system of differential equations a generator
X = ␰i共x,u兲
⳵
⳵
+ ␩␣共x,u兲 ␣
⳵xi
⳵u
共3兲
of a continuous transformation group admitted by differential equations under consideration.
A vector field C = 共C1 , . . . , Cn兲 is said to be a conserved vector for the differential equations 共2兲
if the equation
Di共Ci兲 = 0
共4兲
holds for any solution of Eq. 共2兲.
If one of the independent variables is time, e.g., xn = t, then the conservation law is often
written in the form
dE
= 0,
dt
where
E=
冕
Rn−1
Cn共x,u共x兲,u共1兲共x兲兲dx1 ¯ dxn−1 .
共5兲
Accordingly, Cn is termed the density of the conservation law.
B. Basic conservation theorem
We will employ the recent general theorem 共Ibragimov, 2006兲 on a connection between
symmetries and conservation laws for arbitrary systems of sth-order partial differential equations
F␣共x,u,u共1兲, . . . ,u共s兲兲 = 0,
␣ = 1, . . . ,m,
共6兲
where F␣共x , u , u共1兲 , . . . , u共s兲兲 involves n independent variables x = 共x , . . . , x 兲 and m dependent
variables u = 共u1 , . . . , um兲, u = u共x兲 together with their derivatives up to an arbitrary order s. For our
purposes, we formulate the theorem in the case of systems of first-order differential equations.
Theorem 2.1: 共See Ibragimov, 2006, Theorem 3.5兲. Let an operator 共3兲 be a symmetry of a
system of first-order partial differential equations,
1
F␣共x,u,u共1兲兲 = 0,
␣ = 1, . . . ,m,
where v = 共v1 , . . . , vm兲. Then the quantities
冋
Ci = v␤ ␰iF␤ + 共␩␣ − ␰ ju␣j 兲
册
⳵F␤
,
⳵u␣i
i = 1, . . . ,n
n
共7兲
共8兲
furnish a conserved vector C = 共C1 , . . . , Cn兲 for the equations 共7兲 considered together with the
adjoint system
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053523-3
J. Math. Phys. 48, 053523 共2007兲
Conservation laws for the Maxwell-Dirac equations
F␣* 共x,u, v,u共1兲, v共1兲兲 ⬅
␦L
= 0,
␦u␣
␣ = 1, . . . ,m,
共9兲
where
⳵
␦
⳵
=
− Di ␣ ,
␦u␣ ⳵u␣
⳵ui
␣ = 1, . . . ,m,
and v = 共v1 , . . . , vm兲 are new dependent variables, i.e., v = v共x兲.
Remark 2.1: The simultaneous system of Eqs. 共7兲 and 共9兲 with 2m dependent variables u
= 共u1 , . . . , um兲, v = 共v1 , . . . , vm兲 can be obtained as the Euler-Lagrange equations 共2兲 with the Lagrangian
L = v␤F␤共x,u,u共1兲, . . . ,u共s兲兲.
共10兲
␦L
= F␣共x,u,u共1兲兲,
␦v␣
共11兲
␦L
= F␣* 共x,u, v,u共1兲, v共1兲兲.
␦u␣
共12兲
Indeed,
Remark 2.2: The conserved quantities 共8兲 can be written in terms of the Lagrangian 共10兲 as
follows:
Ci = L␰i + 共␩␣ − ␰ ju␣j 兲
⳵L
⳵u␣i
共13兲
.
Remark 2.3: If Eqs. 共7兲 have r symmetries X1 , . . . , Xr of the form 共3兲,
X␮ = ␰␮i 共x,u兲
⳵
⳵
+ ␩␮␣共x,u兲 ␣ ,
⳵xi
⳵u
␮ = 1, . . . ,r,
then Eqs. 共8兲 provide r conserved vectors C1 , . . . , Cr with the components
C␮i = L␰␮i + 共␩␮␣ − ␰␮j u␣j 兲
⳵L
⳵u␣i
,
␮ = 1, . . . ,r,
i = 1, . . . ,n.
III. ELECTROMAGNETIC EQUATIONS
A. Basic equations and the Lagrangian
Adopting Dirac’s ideas on the existence of magnetic monopoles 共Dirac, 1931兲, one can formulate a symmetrized version of Maxwell’s electromagnetic equations 共Schwinger, 1969兲. In
Systeme International 共SI兲 units and in microscopic 共Lorentz兲 form, these equations are 关cf. Thidé,
2006, Eqs. 共1.50兲兴
⳵B
+ ␮0jm = 0,
⳵t
共14a兲
1 ⳵E
− ␮0je = 0,
c2 ⳵t
共14b兲
⵱⫻E+
⵱⫻B−
⵱ · E − ␮0c2␳e = 0,
共14c兲
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053523-4
J. Math. Phys. 48, 053523 共2007兲
Ibragimov, Khamitova, and Thidé
⵱ · B − ␮0␳m = 0,
共14d兲
together with dual Ohm’s law
je = ␴eE,
jm = ␴mB,
共15兲
where ␴m and ␴e are constant scalar 共rank zero兲 quantities. The first equation in Eq. 共15兲 is Ohm’s
law for electric currents. The second equation is dual Ohm’s law for magnetic currents, that, for
symmetry reasons, was introduced by Thidé, 2006, Eqs. 共2.60兲; see also Eq. 共5兲 by Meyer-Vernet,
1982, Eq. 共38兲 by Olesen, 1996, and its generalization by Coceal et al., 1996.
Now we substitute Eqs. 共15兲 into Eqs. 共14a兲 and 共14b兲. The ensuing equations involve, along
with the light velocity c, three other constants, ␴e, ␴m, and ␮0. We eliminate two constants by
setting
t̃ = ct,
B̃ = cB,
˜␴e = c␮0␴e,
˜␳e = c2␮0␳e,
˜␴m =
␮0
␴m ,
c
˜␳m = c␮0␳m ,
共16兲
and rewrite our basic Maxwell-Dirac equations 共14兲, discarding tilde, as follows:
⵱⫻E+
⳵B
+ ␴mB = 0,
⳵t
⵱⫻B−
⳵E
− ␴eE = 0,
⳵t
⵱ · E − ␳e = 0,
⵱ · B − ␳m = 0.
共17兲
The system 共17兲 has eight equations for eight dependent variables: six coordinates of the electric
and magnetic vector fields E = 共E1 , E2 , E3兲 and B = 共B1 , B2 , B3兲, respectively, and two scalar quantities, viz., the electric and magnetic monopole charge densities ␳e and ␳m.
Using the method of Ibragimov 共2006兲 we write the Lagrangian 共10兲 for Eqs. 共17兲 in the
following form:
冉
L=V· ⵱⫻E+
冊
冉
冊
⳵B
⳵E
+ ␴mB + Re共⵱ · E − ␳e兲 + W · ⵱ ⫻ B −
− ␴eE + Rm共⵱ · B − ␳m兲,
⳵t
⳵t
共18兲
where V , W , Re , Rm are adjoint variables 共we note in passing that V is a pseudovector and Rm a
pseudoscalar兲. With this Lagrangian we have
␦L
⳵B
=⵱⫻E+
+ ␴mB,
␦V
⳵t
␦L
= ⵱ · E − ␳e ,
␦Re
␦L
⳵E
=⵱⫻B−
− ␴eE,
␦W
⳵t
␦L
= ⵱ · B − ␳m ,
␦Rm
共19兲
and
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053523-5
J. Math. Phys. 48, 053523 共2007兲
Conservation laws for the Maxwell-Dirac equations
␦L
⳵W
=⵱⫻V+
− ␴eW − ⵱Re,
␦E
⳵t
␦L
= − Re ,
␦␳e
␦L
⳵V
=⵱⫻W−
+ ␴mV − ⵱Rm,
␦B
⳵t
␦L
= − Rm .
␦␳m
共20兲
It follows from Eqs. 共19兲 and 共20兲 that the Euler-Lagrange equations 共2兲 for the Lagrangian 共18兲
provide the electromagnetic equations 共17兲 and the following adjoint equations for the new dependent variables V , W , Re , Rm:
⵱⫻V+
⳵W
− ␴eW = 0,
⳵t
⵱⫻W−
⳵V
+ ␴mV = 0,
⳵t
Re = 0,
共21兲
Rm = 0.
1
2
3
Remark 3.1: Let the spatial coordinates x , x , x be x , y , z. For computing the variational
derivatives ␦L / ␦E and ␦L / ␦B in Eqs. 共20兲, it is convenient to use the coordinate representation of
the Lagrangian 共18兲, namely,
L = V1共E3y − Ez2 + B1t + ␴mB1兲 + V2共Ez1 − E3x + B2t + ␴mB2兲 + V3共E2x − E1y + B3t + ␴mB3兲 + Re共E1x + E2y
+ Ez3 − ␳e兲 + W1共B3y − Bz2 − E1t − ␴eE1兲 + W2共Bz1 − B3x − E2t − ␴eE2兲 + W3共B2x − B1y − E3t − ␴eE3兲
+ Rm共B1x + B2y + Bz3 − ␳m兲.
共22兲
B. Symmetries
Equations 共17兲 are invariant under the translations of time t and the position vector x
= 共x , y , z兲 as well as the simultaneous rotations of the vectors x, E, and B due to the vector
formulation of Eqs. 共17兲. These geometric transformations provide the following seven infinitesimal symmetries:
X0 =
⳵
,
⳵t
X1 =
⳵
,
⳵x
X2 =
⳵
,
⳵y
X3 =
⳵
,
⳵z
X12 = y
⳵
⳵
⳵
⳵
⳵
⳵
− x + E2 1 − E1 2 + B2 1 − B1 2 ,
⳵x
⳵y
⳵E
⳵E
⳵B
⳵B
X13 = z
⳵
⳵
⳵
⳵
⳵
⳵
− x + E3 1 − E1 3 + B3 1 − B1 3 ,
⳵x
⳵z
⳵E
⳵E
⳵B
⳵B
X23 = z
⳵
⳵
⳵
⳵
⳵
⳵
− y + E3 2 − E2 3 + B3 2 − B2 3 .
⳵y
⳵z
⳵E
⳵E
⳵B
⳵B
共23兲
The infinitesimal symmetries for the adjoint system 共21兲 are obtained from Eq. 共23兲 by replacing
the vectors E and B by V and W, respectively. Moreover, since Eqs. 共17兲 are homogeneous, they
admit simultaneous dilations of all dependent variables with the generator
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053523-6
J. Math. Phys. 48, 053523 共2007兲
Ibragimov, Khamitova, and Thidé
T=E·
⳵
⳵
⳵
⳵
+B·
+ ␳e
+ ␳m
,
⳵␳e
⳵␳m
⳵E
⳵B
共24兲
where
3
E·
⳵
⳵
= 兺 Ei ,
⳵E i=1 ⳵Ei
3
B·
⳵
⳵
= 兺 Bi
.
⳵B i=1 ⳵Bi
Recall that the Maxwell equations in vacuum admit also the one-parameter group of
Heaviside-Larmor-Rainich duality transformations
Ē = E cos ␣ − B sin ␣,
B̄ = E sin ␣ + B cos ␣
共25兲
with the generator
3
X=E·
冉
冊
⳵
⳵
⳵
⳵
−B·
⬅ 兺 Ei i − Bi i .
⳵B
⳵E
⳵B
⳵E i=1
Also recall that the “mixing angle” ␣ in Eq. 共25兲 is a pseudoscalar.
It was shown 共Ibragimov, 2006兲 that the group 共25兲 provides the conservation of energy for
the Maxwell equations. Let us clarify whether Eqs. 共17兲 admit a similar group. Let therefore
X=E·
⳵
⳵
⳵
⳵
− ␳m
.
−B·
+ ␳e
⳵␳m
⳵␳e
⳵B
⳵E
共26兲
The prolongation of the operator 共26兲 is written
X=E·
⳵
⳵
⳵
⳵
⳵
⳵
⳵
⳵
⳵
− ␳m
+ Et ·
− Bt ·
+ Ex ·
− Bx ·
+ Ey ·
−B·
+ ␳e
⳵␳m
⳵␳e
⳵B
⳵E
⳵Bt
⳵Et
⳵Bx
⳵Ex
⳵B y
− By ·
⳵
⳵
⳵
+ Ez ·
− Bz ·
.
⳵E y
⳵Bz
⳵Ez
共27兲
Reckoning shows that the operator 共27兲 acts on the left-hand sides of Eqs. 共17兲 as follows:
X共⵱ ⫻ E + Bt + ␴mB兲 = − 共⵱ ⫻ B − Et − ␴mE兲,
X共⵱ ⫻ B − Et − ␴eE兲 = ⵱ ⫻ E + Bt + ␴eB,
X共⵱ · E − ␳e兲 = − 共⵱ · B − ␳m兲,
X共⵱ · B − ␳m兲 = ⵱ · E − ␳e .
It follows that the operator 共26兲 is admitted by Eqs. 共17兲 only in the case
␴m = ␴e .
共28兲
IV. CONSERVATION LAWS
A. Derivation of conservation laws
We will write the conservation law 共4兲 in the form
Dt共␶兲 + div ␹ = 0,
共29兲
where the pseudoscalar ␶ is the density of the conservation law 共29兲, the pseudovector current
␹ = 共␹1 , ␹2 , ␹3兲, and
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053523-7
J. Math. Phys. 48, 053523 共2007兲
Conservation laws for the Maxwell-Dirac equations
div ␹ ⬅ ⵱ · ␹ = Dx共␹1兲 + Dy共␹2兲 + Dz共␹3兲.
Let us find the conservation law furnished by the symmetry 共26兲 when the condition 共28兲 is
satisfied, ␴m = ␴e. Applying the formula 共8兲 to the symmetry 共26兲 and to the Lagrangian 共18兲, we
obtain the following density of the conservation law 共29兲:
␶=E·
⳵L
⳵Bt
−B·
⳵L
⳵Et
= E · V + B · W.
Thus,
␶ = E · V + B · W.
共30兲
The pseudovector ␹ is obtained likewise. For example, using the Lagrangian in the form 共22兲, we
have
␹1 = E ·
⳵L
⳵Bx
−B·
⳵L
⳵Ex
= E 2W 3 − E 3W 2 − B 2V 3 + B 3V 2 .
The other coordinates of ␹ are computed likewise, and the final result is
␹ = 共E ⫻ W兲 − 共B ⫻ V兲.
共31兲
One can readily verify that Eqs. 共30兲 and 共31兲 provide a conservation law for Eqs. 共17兲 considered
together with the adjoint equations 共21兲. Indeed, using the well-known formula ⵱ · 共a ⫻ b兲
= b · 共⵱ ⫻ a兲 − a · 共⵱ ⫻ b兲 and Eqs. 共17兲 and 共21兲, we obtain
Dt共␶兲 = Et · V + E · Vt + Bt · W + B · Wt = V · 共⵱ ⫻ B − ␴eE兲 + E · 共⵱ ⫻ W + ␴mV兲
− W共⵱ ⫻ E + ␴mB兲 − B · 共⵱ ⫻ V − ␴eW兲,
⵱ · ␹ = ⵱ · 共E ⫻ W兲 − 共 ⵱ · B ⫻ V兲 = W · 共⵱ ⫻ E兲 − E · 共⵱ ⫻ W兲 − V · 共⵱ ⫻ B兲 + B · 共⵱ ⫻ V兲.
Whence,
Dt共␶兲 + div ␹ = 共␴m − ␴e兲共E · V − B · W兲.
It follows again that the conservation law is valid only if ␴m = ␴e.
Remark 4.1: The conservation law given by Eqs. 共30兲 and 共31兲 depends on solutions 共V , W兲
of the adjoint system 共21兲. However, substituting into Eqs. 共30兲 and 共31兲 any particular solution
共V , W兲 of the adjoint system 共21兲 with ␴m = ␴e, one obtains the conservation law for Eqs. 共17兲 not
involving V and W. Let us denote ␴m = ␴e = ␴ and take, e.g., the following simple solution of the
adjoint system 共21兲:
V 1 = e ␴t,
V2 = V3 = 0,
W 1 = e ␴t,
W2 = W3 = 0.
Then Eqs. 共30兲 and 共31兲 yield
␶ = 共E1 + B1兲e␴t ,
␹1 = 0,
␹2 = 共E3 − B3兲e␴t,
␹3 = 共B2 − E2兲e␴t .
Remark 4.2: The operator 共26兲 generates the one-parameter group
Ē = E cos ␣ − B sin ␣,
B̄ = E sin ␣ + B cos ␣ ,
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053523-8
J. Math. Phys. 48, 053523 共2007兲
Ibragimov, Khamitova, and Thidé
¯␳e = ␳e cos ␣ − ␳m sin ␣,
¯␳m = ␳e sin ␣ + ␳m cos ␣ .
where, again, the “mixing angle” ␣ is a pseudoscalar.
Remark 4.3: In the original variables used in Eqs. 共14a兲, 共14b兲, and 共15兲, the operator 共26兲 is
written as
1
1
⳵
⳵
⳵
⳵
X= E·
− cB ·
+ c␳e
− ␳m
.
⳵␳m c ⳵␳e
c
⳵B
⳵E
Applying similar calculations to the generator 共24兲 of the dilation group provides the conservation law with
␶ = B · V − E · W,
␹ = 共E ⫻ V兲 + 共B ⫻ W兲.
共32兲
This conservation law is valid for arbitrary ␴m and ␴e. Indeed,
Dt共␶兲 = Bt · V + B · Vt − Et · W − E · Wt = − V · 共⵱ ⫻ E + ␴mB兲 + B · 共⵱ ⫻ W + ␴mV兲
− W · 共⵱ ⫻ B − ␴eE兲 + E · 共⵱ ⫻ V − ␴eW兲 = − V · 共⵱ ⫻ E兲 + B · 共⵱ ⫻ W兲
− W · 共⵱ ⫻ B兲 + E · 共⵱ ⫻ V兲,
⵱ · ␹ = ⵱ · 共E ⫻ V兲 + ⵱ · 共B ⫻ W兲 = V · 共⵱ ⫻ E兲 − E · 共⵱ ⫻ V兲 + W · 共⵱ ⫻ B兲 − B · 共⵱ ⫻ W兲.
Hence, Dt共␶兲 + div ␹ = 0.
Let us find the conservation law provided by the symmetry X0 = ⳵ / ⳵t from Eq. 共23兲. Formula
共8兲 yields
␶ = L − Et ·
⳵L
⳵Et
⳵L
− Bt ·
⳵Bt
= L + Et · W − Bt · V.
Since the Lagrangian L given by Eq. 共18兲 vanishes on the solutions of Eqs. 共17兲, we can take
␶ = Et · W − Bt · V
共33兲
␶ = W · 关共⵱ ⫻ B兲 − ␴eE兴 + V · 关共⵱ ⫻ E兲 + ␴mB兴.
共34兲
or
Let us calculate the pseudovector ␹. Formula 共8兲 yields
␹1 = − Et ·
⳵L
⳵Ex
− Bt ·
⳵L
⳵Bx
.
Using the Lagrangian in the form 共22兲, we have
␹1 = − E2t V3 + E3t V2 − B2t W3 + B3t W2 .
The other coordinates of ␹ are computed similarly, and the final result is
␹ = 共V ⫻ Et兲 + 共W ⫻ Bt兲.
共35兲
Thus, the time translational invariance of Eqs. 共17兲 leads to the conservation law 共29兲 with ␶ and
␹ given by Eqs. 共34兲 and 共35兲, respectively.
Remark 4.4: Let us substitute in Eqs. 共34兲 and 共35兲 the following simple solution of the
adjoint system 共cf. Remark 4.1兲:
V 1 = e ␴mt,
V2 = V3 = 0,
W 1 = e ␴et,
W2 = W3 = 0.
Then Eqs. 共34兲 and 共35兲 yield
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053523-9
J. Math. Phys. 48, 053523 共2007兲
Conservation laws for the Maxwell-Dirac equations
␶ = 共B3y − Bz2 − ␴eE1兲e␴et + 共E3y − Ez2 + ␴mB1兲e␴mt ,
␹1 = 0,
␹2 = − E3t e␴mt − B3t e␴et,
␹3 = E2t e␴mt + B2t e␴et .
The conservation law provided by the symmetry X1 = ⳵ / ⳵x from Eq. 共23兲 has the following
density:
␶ = − Ex ·
⳵L
⳵Et
− Bx ·
⳵L
⳵Bt
= Ex · W − Bx · V.
For the pseudovector ␹ the formula 共8兲 yields
␹1 = L − Ex ·
⳵L
⳵Ex
− Bx ·
⳵L
⳵Bx
.
Using the Lagrangian in the form 共22兲, we have
␹1 = L − E2x V3 + E3x V2 − B2x W3 + B3x W2 .
The other coordinates of ␹ are calculated similarly:
␹2 = E1x V3 − E3x V1 + B1x W3 − B3x W1 ,
␹3 = − E1x V2 + E2x V1 − B1x W2 + B2x W1 .
We can ignore L in ␹ since DxL = 0 on solutions of Eqs. 共17兲 and 共21兲, and the final result is
1
␹ = 共V ⫻ Ex兲 + 共W ⫻ Bx兲.
共36兲
Replacing x by y and z we obtain the following conservation laws corresponding to X2 = ⳵ / ⳵y and
X3 = ⳵ / ⳵z, respectively:
␶ = Ey · W − By · V,
␹ = 共V ⫻ Ey兲 + 共W ⫻ By兲
␶ = Ez · W − Bz · V,
␹ = 共V ⫻ Ez兲 + 共W ⫻ Bz兲.
and
Applying formula 共8兲 to the symmetry X12 and to the Lagrangian 共22兲, we obtain the following
density of the conservation law:
␶ = E2
⳵L
⳵E1t
− E1
⳵L
⳵E2t
+ 共xEy − yEx兲 ·
⳵L
⳵Et
+ B2
⳵L
⳵B1t
− B1
⳵L
⳵B2t
+ 共xBy − yBx兲 ·
⳵L
⳵Bt
= W 2E 1 − W 1E 2
+ 共yEx − xEy兲 · W − 共V2B1 − V1B2兲 − 共yBx − xBy兲 · V.
The densities of the conservation laws for X13 and X23 are
␶ = W3E1 − W1E3 + 共zEx − xEz兲 · W − 共V3B1 − V1B3兲 − 共zBx − xBz兲 · V
and
␶ = W3E2 − W2E3 + 共zEy − yEz兲 · W − 共V3B2 − V2B3兲 − 共zBy − yBz兲 · V,
respectively. Finally, the densities of conservation laws corresponding to the rotation generators
Xij can be written as one vector:
␶ = W ⫻ E + W · 共x ⫻ ⵱兲E − V ⫻ B − V · 共x ⫻ ⵱兲B,
共37兲
where x = 共x , y , z兲.
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053523-10
Ibragimov, Khamitova, and Thidé
J. Math. Phys. 48, 053523 共2007兲
The operator X12 provides the following pseudovector ␹:
␹1 = − V3E1 − y共E2x V3 − E3x V2兲 + x共E2y V3 − E3y V2兲 − W3B1 − y共B2x W3 − B3x W2兲 + x共B2y W3 − B3y W2兲,
␹2 = − V3E2 + y共E1x V3 − E3x V1兲 − x共E1y V3 − E3y V1兲 − W3B2 + y共B1x W3 − B3x W1兲 − x共B1y W3 − B3y W1兲,
␹3 = V1E1 + V2E2 − y共E1x V2 − E2x V1兲 + x共E1y V2 − E2y V1兲 + W1B1 + W2B2 − y共B1x W2 − B2x W1兲 + x共B1y W2
− B2y W1兲.
The pseudovector ␹ for the operator X13 has the following form:
␹1 = V2E1 − z共E2x V3 − E3x V2兲 + x共Ez2V3 − Ez3V2兲 + W2B1 − z共B2x W3 − B3x W2兲 + x共Bz2W3 − Bz3W2兲,
␹2 = − V1E1 − V3E3 + z共E1x V3 − E3x V1兲 − x共Ez1V3 − Ez3V1兲 − W1B1 − W3B3 + z共B1x W3 − B3x W1兲
− x共Bz1W3 − Bz3W1兲,
␹3 = V2E3 − z共E1x V2 − E2x V1兲 + x共Ez1V2 − Ez2V1兲 + W2B3 − z共B1x W2 − B2x W1兲 + x共Bz1W2 − Bz2W1兲.
The operator X23 provides the following pseudovector ␹:
␹1 = V2E2 + V3E3 − z共E2y V3 − E3y V2兲 + y共Ez2V3 − Ez3V2兲 + W2B2 + W3B3 − z共B2y W3 − B3y W2兲 + y共Bz2W3
− Bz3W2兲,
␹2 = − V1E2 + z共E1y V3 − E3y V1兲 − y共Ez1V3 − Ez3V1兲 − W1B2 + z共B1y W3 − B3y W1兲 − y共Bz1W3 − Bz3W1兲,
␹3 = − V1E3 − z共E1y V2 − E2y V1兲 + y共Ez1V2 − Ez2V1兲 − W1B3 − z共B1y W2 − B2y W1兲 + y共Bz1W2 − Bz2W1兲.
B. Two-solution representation of conservation laws
The conserved quantities obtained in Sec. IV A involve solutions V , W of the adjoint equations 共21兲. It may be useful for applications to give an alternative representation of the conserved
quantities in terms of the electric and magnetic vector fields E , B only.
We suggest here one possibility based on the observation that one can satisfy the adjoint
system 共21兲 by letting
V共x,t兲 = B共x,− t兲,
W共x,t兲 = E共x,− t兲,
Re共x,t兲 = ⵱ · E共x,− t兲 − ␳e共x,− t兲,
Rm共x,t兲 = ⵱ · B共x,− t兲 − ␳m共x,− t兲,
共38兲
where E共x , s兲 , B共x , s兲 solve Eqs. 共17兲 with s = −t. Indeed, employing the substitution 共38兲 and the
notation s = −t we have
⵱⫻V+
⳵E共x,s兲 ⳵s
⳵W
− ␴eE共x,s兲,
− ␴eW = ⵱ ⫻ B共x,s兲 +
⳵t
⳵s ⳵t
⵱⫻W−
⳵B共x,s兲 ⳵s
⳵V
+ ␴mB共x,s兲,
+ ␴mV = ⵱ ⫻ E共x,s兲 −
⳵t
⳵s ⳵t
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053523-11
Conservation laws for the Maxwell-Dirac equations
J. Math. Phys. 48, 053523 共2007兲
Re = ⵱ · E共x,s兲 − ␳e共x,s兲,
Rm = ⵱ · B共x,s兲 − ␳m共x,s兲.
共39兲
Hence, the adjoint Eqs. 共21兲 reduce to Eq. 共17兲:
⵱ ⫻ E共x,s兲 +
⳵B共x,s兲
+ ␴mB共x,s兲 = 0,
⳵s
⵱ · B共x,s兲 −
⳵E共x,s兲
− ␴eE共x,s兲 = 0,
⳵s
⵱ · E共x,s兲 − ␳e共x,s兲 = 0,
⵱ · B共x,s兲 − ␳m共x,s兲 = 0.
共40兲
Let 共E共x , t兲 , B共x , t兲兲 and 共E⬘共x , t兲 , B⬘共x , t兲兲 be any two solutions of the electromagnetic equations 共17兲. Substituting in Eq. 共38兲 the solution 共E⬘ , B⬘兲, we obtain the two-solution representations of the conservation laws. For example, the conservation law given by Eqs. 共30兲 and 共31兲 has
in this representation the following coordinates:
␶ = E共x,t兲 · B⬘共x,− t兲 + B共x,t兲 · E⬘共x,− t兲,
␹ = 关E共x,t兲 ⫻ E⬘共x,− t兲兴 − 关B共x,t兲 ⫻ B⬘共x,− t兲兴.
共41兲
In particular, if the solutions 共E共x , t兲 , B共x , t兲兲 are identical, Eq. 共41兲 provides the one-solution
representation:
␶ = E共x,t兲 · B共x,− t兲 + B共x,t兲 · E共x,− t兲,
␹ = 关E共x,t兲 ⫻ E共x,− t兲兴 − 关B共x,t兲 ⫻ B共x,− t兲兴.
共42兲
All other conservation laws can be treated likewise, e.g., the conservation law given by Eqs.
共33兲 and 共35兲 has the following two-solution representation:
␶ = Et共x,t兲 · E⬘共x,− t兲 − Bt共x,t兲 · B⬘共x,− t兲,
␹ = 关B⬘共x,− t兲 ⫻ Et共x,t兲兴 + 关E⬘共x,− t兲 ⫻ Bt共x,t兲兴.
共43兲
ACKNOWLEDGMENT
One of the authors 共B.T.兲 gratefully acknowledges the financial support from the Swedish
Governmental Agency for Innovation Systems 共VINNOVA兲.
N. H. Ibragimov, J. Math. Anal. Appl. 333, 311–320 共2007兲.
N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, 2nd ed. 共Wiley, Chichester,
1999兲.
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P. A. M. Dirac, Proc. R. Soc. London, Ser. A 133, 60 共1931兲.
4
J. Schwinger, Science 165, 757 共1969兲.
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B. Thidé, Electromagnetic Field Theory 共Upsilon Books, Uppsala, Sweden, 2006兲; http://www.plasma.uu.se/CED/Book
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N. Meyer-Vernet, Am. J. Phys. 50, 846 共1982兲.
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Paper IV
59
ARTICLE IN PRESS
Commun Nonlinear Sci Numer Simulat xxx (2008) xxx–xxx
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat
journal homepage: www.elsevier.com/locate/cnsns
Symmetries and nonlocal conservation laws of the general magma equation
Raisa Khamitova *
Department of Mathematics and Science, Blekinge Institute of Technology, SE-371 79 Karlskrona, Sweden
a r t i c l e
i n f o
Article history:
Received 22 August 2008
Accepted 22 August 2008
Available online xxxx
PACS:
02.30.Jr
02.20.Sv
Keywords:
Magma equation
Self-adjointness
Quasi-self-adjointness
Nonlocal conservation laws
a b s t r a c t
In this paper the general magma equation modelling a melt flow in the Earth’s mantle is
discussed. Applying the new theorem on nonlocal conservation laws [Ibragimov NH. A
new conservation theorem. J Math Anal Appl 2007;333(1):311–28] and using the symmetries of the model equation nonlocal conservation laws are computed. In accordance
with Ibragimov [Ibragimov NH. Quasi-self-adjoint differential equations. Preprint in
Archives of ALGA, vol. 4, BTH, Karlskrona, Sweden: Alga Publications; 2007. p. 55–60,
ISSN: 1652-4934] it is shown that the general magma equation is quasi-self-adjoint for
arbitrary m and n and self-adjoint for n = m. These important properties are used for
deriving local conservation laws.
Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction
The equation
of
o
o 1 of
¼0
þ
fn 1 ot oz
oz f m ot
ð1Þ
models the migration of melt through the Earth’s mantle. In [3] it is called the general magma equation. It follows from the
equations
uz ¼ ft ;
o 1 ou
u ¼ fn 1 þ
;
oz f m oz
ð2Þ
where u is the vertical barometric flux of melt, f is the volume fraction of melt, z is a vertical space coordinate and t is time.
All the variables are dimensionless. Eqs. (2) were proposed by Scott and Stevenson [4]. They suggested that according to Dullien [5] 2 6 n 6 5 or even bigger and supposed that 0 6 m 6 1. Some authors discussed Eq. (1) for any values of n and m.
The general magma equation does not have any Lagrangian and therefore it is impossible to apply Noether’s theorem for
calculating conservation laws. However, according to Atherton and Homsy [6] and Ibragimov [1], it is possible to introduce a
formal Lagrangian if any given system of equations is taken into consideration together with the adjoint system. In his recent
paper [1], Ibragimov has proved that the adjoint system inherits symmetries of the given system and has suggested a new
theorem on nonlocal conservation laws. The nonlocal conserved vectors presented in my article are obtained by applying this
new theorem to the infinitesimal symmetries of Eq. (1). Preliminary results were published in the preprint [7] and in the
proceedings of the 2nd conference on ‘‘Nonlinear Science and Complexity” [8].
* Tel.: +46 455 385 462.
E-mail address: [email protected]
1007-5704/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.cnsns.2008.08.009
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I denote f by u in order to make Eq. (1) compatible with the usual notation in the Lie group analysis. Using the notation of
differential algebra it is rewritten as follows:
F ut þ Dz fun ½1 Dz ðum ut Þg ¼ 0
ð3Þ
where
Dz ¼
o
o
o
o
þ uzjk
þ þ uz
þ uzj
oz
ou
ouj
oujk
is the operator of total differentiation with respect to z. Here the usual convention of summation over repeated indices is
used. Correspondingly, Dt is the operator of total differentiation with respect to t.
The local conservation laws for Eq. (3),
½Dt ðC 1 Þ þ Dz ðC 2 Þð3Þ ¼ 0;
were obtained by Barcilon and Richter [9] and Harris [10] using the direct method of calculating components of conserved
vectors. These conservation laws were also discussed in [11]. It is shown that the local conserved vectors of the form
C 1 ¼ C 1 ðu; uz Þ and C 2 ¼ Aðu; uz Þut þ Bðu; uz Þutz þ Sðu; uz Þ;
which are presented in [11] as Case B in Table 2 can be calculated by using self-adjointness and quasi-self-adjointness of Eq.
(3). The local conserved vector with the components
e 2 ¼ munm1 uz ut u12m utz þ
C
e 1 ¼ u;
C
n
u1m
1m
ð4Þ
can be derived as a particular case of a nonlocal conservation law (see [11], Case A in Table 2).
2. Self-adjointness of the equation
After differentiations Eq. (3) transforms to
F ut unm utzz þ ð2m nÞunm1 uz utz þ munm1 ut uzz þ mðn m 1Þunm2 ut u2z þ nun1 uz ¼ 0:
ð5Þ
The adjoint equation F* = 0 is defined according to [1]
F d
ðvFÞ ¼ 0
du
ð6Þ
where v = v(t, z) is a new dependent variable and
d
o
o
o
o
þ Di Dj
Di Dj Dk
þ ¼
Di
du ou
oui
ouij
ouijk
is the variational derivative, D1 = Dt and D2 = Dz, i,j,k = 1,2. Eqs. (5) and (6) yield
F vt þ vtzz unm þ nvtz unm1 uz þ nvzz unm1 ut þ nvz ½2unm1 utz þ ðn m 1Þunm2 ut uz un1 ¼ 0:
ð7Þ
Letting v = u in Eq. (7) and comparing it with Eq. (5) we obtain that F = F = 0 if m = n. We have proved the following:
*
Proposition 2.1. Eq. (5) is self-adjoint if m = n, i.e. when it has the following form:
F ut u2n utzz 3nu2n1 uz utz nu2n1 ut uzz nð2n 1Þu2n2 ut u2z þ nun1 uz ¼ 0:
ð8Þ
3. Quasi-self-adjointness of the equation
According to Ibragimov [2] an equation is quasi-self-adjoint if there exists a function h(u) such that F* = k(u)F upon the
substitution v = h(u).
Proposition 3.1. Eq. (5) is quasi-self-adjoint for arbitrary m and n.
Proof. Calculation of the derivatives of v = h(u):
0
vt ¼ h ut ;
0
00
vz ¼ h uz ;
0
vtz ¼ h ut uz þ h utz ;
00
0
vzz ¼ h u2z þ h uzz ;
000
00
00
0
vtzz ¼ h ut u2z þ 2h uz utz þ h ut uzz þ h utzz
and substitution of the results in (7) give
0
00
0
00
0
000
00
F ¼ h ðut utzz unm Þ þ ð2uh þ 3nh Þunm1 uz utz þ ðuh þ nh Þunm1 ut uzz þ ½u2 h þ 2nuh
0
0
þ nðn m 1Þh unm2 ut u2z h nuuz :
ð9Þ
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0
Comparing with Eq. (5) we obtain k(u) = h and
00
0
0
2uh þ 3nh ¼ ð2m nÞh ;
2 000
00
00
0
0
uh þ nh ¼ mh ;
0
ð10Þ
0
u h þ 2nuh þ nðn m 1Þh ¼ mðn m 1Þh :
ð11Þ
Eqs. (10) reduce to one equation, namely:
00
0
uh þ ðn þ mÞh ¼ 0:
The latter equation has two solutions:
h¼
C1
þ C2;
ðn þ m 1Þunþm1
when n þ m–1
and
h ¼ C 1 ln juj þ C 2 ;
when n þ m ¼ 1;
C1 and C2 are arbitrary constants. These functions also satisfy Eq. (11). Thus Eq. (5) is quasi-self-adjoint. We can choose
hðuÞ ¼ u1nm
ð12Þ
hðuÞ ¼ ln juj
ð13Þ
and
for n + m – 1 and n + m = 1, respectively.
h
4. Nonlocal conservation laws: general form
In accordance with [6,1] we introduce the formal Lagrangian
L ¼ vF:
I will write it in the symmetrized form:
1
1
L ¼ v ut unm ðutzz þ uztz þ uzzt Þ þ ð2m nÞunm1 uz ðutz þ uzt Þ þ munm1 ut uzz þ mðn m 1Þunm2 ut u2z þ nun1 uz :
3
2
ð14Þ
Eq. (5) is said to have a nonlocal conservation law if there exits a vector C = (C1, C2) satisfying the condition
Dt ðC 1 Þ þ Dz ðC 2 Þ ¼ 0
ð15Þ
on any solution of Eqs. (5) and (7). Eq. (5) has a local conservation law if (15) is satisfied on any solution of the equation in
question. We will not take into consideration trivial conservation laws.1 Conservation laws are regarded as equivalent if they
differ only by a trivial conservation law.
The conserved vector corresponding to an operator
X ¼ n1 ðt; z; uÞ
o
o
o
þ n2 ðt; z; uÞ þ gðt; z; uÞ ;
ot
oz
ou
admitted by Eq. (5), is obtained by the formula [1]
C i ¼ ni L þ W
oL
oL
oL
oL
oL
oL
þ Dj Dk
þ Dj ðWÞ
þ Dj Dk ðWÞ
Dj
Dk
;
oui
ouij
ouijk
ouij
ouijk
ouijk
ð16Þ
where
W ¼ g ni ui ;
i,j,k = 1, 2. Since the Lagrangian L is equal to zero on solutions of the equation F = 0 we can calculate Ci without the term ni L.
Furthermore, we can rewrite the density
C1 ¼ W
1
oL
oL
oL
oL
oL
oL
Dz
Dz
þ D2z
þ Dz ðWÞ
þ D2z ðWÞ
out
outz
outzz
outz
outzz
outzz
A conservation law is trivial if Dt(C1) + Dz(C2) 0 or C1 and C2 are smooth functions of F, F*, Di(F), Di(F*),. . .
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using the following identities:
oL
oL
oL
WDz
;
¼ Dz W
þ Dz ðWÞ
outz
outz
outz
oL
oL
oL
oL
WD2z
¼ D2z W
2Dz ðWÞDz
D2z ðWÞ
outzz
outzz
outzz
outzz
Hence we obtain
oL
oL
oL
þ Dz ðWÞ 2
3Dz
;
out
outz
outzz
C1 ¼ W
ð17Þ
the remaining term
oL
oL
þ D2z W
Dz W
outz
outzz
ð18Þ
will be included in C2, where it will have the form
Dt ðWÞ
oL
oL
oL
oL
oL
oL
þ Dt Dz ðWÞ
þ Dz ðWÞDt
þ WDt Dz
:
WDt
þ Dt ðWÞDz
outz
outz
outzz
outzz
outzz
outzz
ð19Þ
Calculation of C2 from Eq. (16) gives
oL
oL
oL
oL
oL
oL
oL
oL
Dt
þ Dz ðWÞ
Dt ðWÞDz
Dz
þ Dt Dz
þ Dz Dt
þ Dt ðWÞ
ouz
ouzt
ouzz
ouztz
ouzzt
ouzt
ouzz
ouztz
oL
oL
oL
þ Dz Dt ðWÞ
:
Dz ðWÞDt
þ Dt Dz ðWÞ
ouzzt
ouztz
ouzzt
C2 ¼ W
Using the fact that the Lagrangian (14) is symmetrical with respect to the mixed derivatives and adding the expression (19)
we can simplify C2 and obtain the following:
C2 ¼ W
oL
oL
oL
oL
oL
oL
Dz
þ 3Dt Dz
þ Dz ðWÞ
2Dt
þ 3Dt Dz ðWÞ
:
ouz
outz
ouzz
outzz
ouzz
outzz
ð20Þ
Invoking (14) we get the final expressions for the components of a nonlocal conserved vector from (18) and (20):
C 1 ¼ vfW½1 þ munm1 uzz þ mðn m 1Þunm2 u2z þ ð2m nÞunm1 uz Dz ðWÞg þ Dz ðWÞDz ðvunm Þ
ð21Þ
and
C 2 ¼ Wfv ð2m nÞunm1 utz þ 2mðn m 1Þunm2 ut uz þ nun1 ð2m nÞDt ðvunm1 uz Þ mDz ðvunm1 ut Þ
Dt Dz ðvunm Þg þ mvunm1 ut Dz ðWÞ vunm Dt Dz ðWÞ:
ð22Þ
Thus Eqs. (21) and (22) define components of a nonlocal conservation law for the system of Eqs. (5), (7) corresponding to any
operator X admitted by Eq. (5).
5. Formulation of the results
Here I present the conservation laws obtained in this article. Their computation is given in Sections 6–8.
5.1. Nonlocal conservation laws, n – 0
The symmetries of Eq. (5) for n – 0 provide the following nonlocal conserved vectors:
Translation of time:
C 1 ¼ vz un ;
C 2 ¼ vt ðmunm1 ut uz unm utz þ un Þ þ nvz unm1 u2t þ vtz unm ut ;
Translation of the space coordinate z:
C 1 ¼ ½u þ munm1 u2z unm uzz vz ;
C 2 ¼ vtz unm uz vt u þ nvznm1 ut uz ;
ð23Þ
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o
Dilations with the operator X 3 ¼ ð2 n mÞt oto þ ðn mÞz ozo þ 2u ou
:
C 1 ¼ vz ½ðn mÞzðu þ munm1 u2z unm uzz Þ þ ð2 n mÞðunm uz tun Þ þ ð2 þ n mÞvu;
C 2 ¼ ð2 þ n mÞv½un unm utz þ munm1 ut uz þ vt ½ð2 n mÞt un þ munm1 ut uz unm utz
ðn mÞzu þ nvz ½ð2 n mÞtunm1 u2t 2unm ut þ ðn mÞzunm1 ut uz þ vtz ½ð2 n mÞtunm ut
ð24Þ
2unmþ1 þ ðn mÞzunm uz :
5.2. Local conservation laws, n – 0
Dilations:
Case 1. Eq. (5) is self-adjoint.
n – 0, m = n, m – 1:
C 1 ¼ u2m u2z þ u2 ;
C 2 ¼ 2 mu2m ut uz u12m utz m
u1m :
1m
n = 1, m = 1:
C 1 ¼ u2 u2z þ u2 ;
Case 2a.
Case (i)
C 2 ¼ 2ðu2 ut uz u1 utz ln uÞ:
Eq. (5) is quasi-self-adjoint, n + m – 1:
n + m – 1, n + m – 2, m – 1:
1
1
ð1 n mÞu2m u2z þ
u2nm ;
2
2nm
n
C 2 ¼ mu2m ut uz u12m utz þ
u1m :
1m
C1 ¼
Case (ii)
ð25Þ
n + m = 2, m – 1:
1
C ¼ u2m u2z þ ln u;
2
1
C 2 ¼ mu2m ut uz u12m utz þ
Case (iii)
ð26Þ
ð2 mÞ 1m
u :
1m
n – 1, m = 1:
n 2m 2
1
u uz þ
u1n ;
2
1n
C 2 ¼ u2 ut uz u1 utz þ n ln u:
1
C ¼
Case (iv)
ð27Þ
n = 1, m = 1:
1
C 1 ¼ u2 u2z þ ln u;
2
C 2 ¼ u2 ut uz u1 utz þ ln u:
Case 2b.
Eq. (5) is quasi-self-adjoint, n + m = 1, m – 1:
1 2m 2
u uz þ u ln u
2
2
C ¼ ln u mu2m ut uz u12m utz þ u1m Þ þ mu2m ut uz u12m utz C1 ¼
m
u1m :
1m
5.3. Nonlocal conservation laws, n = 0
When n = 0 Eq. (5) transforms to the form:
1m u
¼ 0;
Dt u D2z
1m
Dt ½u D2z ðln uÞ ¼ 0;
if m–1;
if m ¼ 1:
Translation of the space coordinate z. The conserved vector has the components (23) with n = 0.
e 3 ¼ mz o þ 2u o
Dilations with the operator X
oz
ou
n = 0, m – 0:
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C 1 ¼ vz ½mzðu þ mum1 u2z um uzz Þ þ ð2 mÞum uz þ ð2 mÞvu;
C 2 ¼ v½ð2 mÞum utz þ mð2 mÞum1 ut uz þ mzuvt ½2u1m þ mzum uz vtz :
o
Transformation with the operator X 4 ¼ z2 ozo þ 3zu ou
, n = 0, m ¼ 43:
h
i
4
4
7
C 1 ¼ v 8u3 uz þ z 3u þ 4u3 uzz þ 4u3 u2z
28 10 3
7
4
þ 3u þ 5zuz þ z2 uzz Dz vu3 ;
þz2 uz þ 4u3 uz uzz u 3 uz
9
28 10
4
4
7
C 2 ¼ v u3 ut þ z 3u3 utz 12u3 ut uz z2 ut þ
u 3 ut u2z
9
8 7 4 7 4
Dt vu 3 uz Dz vu 3 ut Dt Dz vu 3
3
3
5.4. Local conservation laws, n = 0
e 3 ¼ mz o þ 2u o
Dilations with the operator X
oz
ou
The equation is quasi-self-adjoint.
Case (i): n = 0, m – 2, m – 1, follows from (25):
1
1
ð1 mÞu2m u2z þ
u2m ;
2
2m
2
2m
12m
utz :
C ¼ mu ut uz u
C1 ¼
Case (ii): n = 0, m = 2, follows from (26):
1 4 2
u uz þ ln u;
2
2
4
C ¼ 2u ut uz u3 utz :
C1 ¼ Case (iii): n = 0, m = 1, follows from (27):
1 2 2
u uz þ u ln u;
2
2
C ¼ ln u½u1 utz þ u2 ut uz þ u2 ut uz u1 utz :
C1 ¼
6. Computation of conservation laws: n 6¼ 0
6.1. Translation of time
Nonlocal conservation law
It is obvious that Eq. (5) with arbitrary m and n is invariant under the translation of time t. The operator for the time translation, X 1 ¼ oto , has n1 = 1, n2 = 0, g = 0, therefore W = ut. Hence from (21) the density C1 has the following form:
C 1 ¼ vfut ½1 þ munm1 uzz þ mðn m 1Þunm2 u2z þ ð2m nÞunm1 uz utz g utz Dz ðvunm Þ:
It follows from Eq. (5), F = 0, that
ut þ ð2m nÞunm1 uz utz þ munm1 ut uzz þ mðn m 1Þunm2 ut u2z ¼ unm utzz nun1 uz
ð28Þ
which gives
C 1 ¼ v½unm utzz þ nun1 uz utz Dz ðvunm Þ ¼ Dz ½vðunm utz þ un Þ vz un :
Hence
C 1 ¼ vz un :
Using (22) and adding the term Dt[v(u
ð29Þ
nm
n
utz + u )] corresponding to Dz[v(u
nm
n
utz + u )] we compute the component C2:
2
C ¼ ut fv½ð2m nÞunm1 utz þ 2mðn m 1Þunm2 ut uz þ nun1 ð2m nÞDt ðvunm1 uz Þ mDz ðvunm1 ut Þ
Dt Dz ðvunm Þg mvunm1 ut utz þ vunm uttz þ Dt ½vðunm utz þ un Þ:
ð30Þ
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The reckoning shows that the expression
v½ð2m nÞunm1 utz þ 2mðn m 1Þunm2 ut uz þ nun1 ð2m nÞDt ðvunm1 uz Þ mDz ðvunm1 ut Þ Dt Dz ðvunm Þ
is equal to
vðnunm1 utz þ nun1 Þ mvt unm1 uz nvz unm1 ut vtz unm :
Hence
C 2 ¼ vt ðmunm1 ut uz unm utz þ un Þ þ nvz unm1 u2t þ vtz unm ut :
ð31Þ
Thus Eqs. (29) and (31) define a nonlocal conservation law corresponding to the translation of the coordinate t. Indeed, calculations give that
Dt ðC 1 Þ þ Dz ðC 2 Þ ¼ vt F þ ut F which equals to zero on solutions of Eqs. (5) and (7).
Local conservation laws:
Eq. (5) is self-adjoint when n = m. Substitution of v = u in (32) yields
C 1 ¼ uz un ¼ Dz
1
unþ1 ;
nþ1
C 1 ¼ uz u1 ¼ Dz ðln uÞ;
if n– 1;
if n ¼ 1:
In the case of quasi-self-adjointness of the equation F = 0, when n + m – 1 and v = u1nm the component C1 has the following
form:
C 1 ¼ ð1 n mÞum uz ¼ Dz
1
1
C ¼ nu uz ¼ Dz ðn ln uÞ;
1 n m 1m
u
;
1m
if m–1;
if m ¼ 1; n–0:
For the second case of quasi-self-adjointness, when n + m = 1 and v = ln u, we have
C 1 ¼ un1 uz ¼ Dz
1 n
u ;
n
C ¼ u1 uz ¼ Dz ðln uÞ;
1
if n–0;
if m ¼ 1;
n ¼ 0:
Thus for arbitrary m and n only trivial local conservation laws correspond to the operator X1.
6.2. Translation of the space coordinate z
Nonlocal conservation law:
Eq. (5) with arbitrary m and n is also invariant under the translation of the coordinate z with the operator X 2 ¼ ozo : In this
case n1 = 0, n2 = 1,g = 0, therefore W = uz. From (21) the corresponding component C1 has the form:
C 1 ¼ vfuz ½1 þ munm1 uzz þ mðn m 1Þunm2 u2z þ ð2m nÞunm1 uz uzz g uzz Dz ðvunm Þ
or
C 1 ¼ ½u þ munm1 u2z unm uzz vz þ Dz ½vðu þ munm1 u2z Þ:
Hence we can choose
C 1 ¼ ½u þ munm1 u2z unm uzz vz
ð32Þ
2
as the density of a nonlocal conservation law corresponding to the translation of z. Adding to C in (22) the term
Dt ½vðu þ munm1 u2z Þ corresponding to Dz ½vðu þ munm1 u2z Þ we obtain:
C 2 ¼ uz fv½ð2m nÞunm1 utz þ 2mðn m 1Þunm2 ut uz þ nun1 ð2m nÞDt ðvunm1 uz Þ mDz ðvunm1 ut Þ
Dt Dz ðvunm Þg mvunm1 ut uzz þ vunm utzz þ Dt ½vðu þ munm1 u2z Þ:
or
C 2 ¼ vtz unm uz vt u þ nvz unm1 ut uz vF:
Excluding the trivial part vF we finally have
C 2 ¼ vtz unm uz vt u þ nvznm1 ut uz :
ð33Þ
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Thus Eqs. (32) and (33) define a nonlocal conservation law corresponding to the translation of the coordinate z. Indeed, calculations give that
Dt ðC 1 Þ þ Dz ðC 2 Þ ¼ vz F þ uz F ;
which equals to zero on solutions of Eqs. (5) and (7).
Local conservation laws:
Self-adjointness of Eq. (5). Substitution of v = u and n = m in (29) gives
C 1 ¼ ½u þ mu2m1 u2z u2m uzz uz ¼ Dz
1 2
ðu u2m u2z Þ :
2
Quasi-self-adjointness of Eq. (5): When n + m – 1 and v = u1nm the component C1 has the following form:
C 1 ¼ ½u þ munm1 u2z unm uzz ð1 n mÞunm uz :
Hence
or
2nm
u
u2m u2z
;
C 1 ¼ Dz ðn þ m 1Þ
þ
nþm2
2
u2m u2z
C 1 ¼ Dz ln u þ
;
2
if n þ m–2
if n þ m ¼ 2:
In the second case of quasi-self-adjointness, when n + m = 1 and v = ln u, we obtain
C 1 ¼ ½u þ mu2m u2z u12m uzz u1 uz ¼ Dz ðu u2m u2z Þ:
Thus for arbitrary m and n only trivial local conservation laws correspond to the operator X2.
6.3. Dilations
Nonlocal conservation law:
When the parameter n – 0, Eq. (5) admits the following operator of dilation [11]:
X 3 ¼ ð2 n mÞt
o
o
o
þ ðn mÞz þ 2u :
ot
oz
ou
For this operator
n1 ¼ ð2 n mÞt;
n2 ¼ ðn mÞz and g ¼ 2u;
therefore
W ¼ 2u ð2 n mÞtut ðn mÞzuz :
From (21) we have the following component C1:
C 1 ¼ vf½2u ð2 n mÞtut ðn mÞzuz ½1 þ munm1 uzz þ mðn m 1Þunm2 u2z þ ð2m nÞunm1 uz Dz ½2u
ð2 n mÞtut ðn mÞzuz g þ ½vz unm þ ðn mÞvunm1 uz ÞDz ½2u ð2 n mÞtut ðn mÞzuz :
Collecting first all terms containing tut and invoking (28) we obtain:
C 11 ¼ ð2 n mÞtfv½nun1 uz unm utzz ðn mÞunm1 uz utz vz unm utz g
¼ ð2 n mÞtun vz þ Dz fð2 n mÞtvðun unm utz Þg:
For the terms containing 2u (n m)zuz we have
C 12 ¼ vz ½ðn mÞzðu þ munm1 u2z unm uzz Þ þ ð2 n mÞunm uz þ ð2 þ n mÞvu þ Dz fv½2munm uz
ðn mÞzðu þ munm1 u2z Þg:
Combining all nontrivial terms in
1
C ¼ vz ðn mÞzðu þ
C 11
and
munm1 u2z
ð34Þ
C 21
u
we obtain the density of a nonlocal conservation law:
nm
uzz Þ þ ð2 n mÞðunm uz tun Þ þ ð2 þ n mÞvu:
ð35Þ
Using (22) and adding the term corresponding to
Dz fv½2munm uz ðn mÞzðu þ munm1 u2z Þ þ ð2 n mÞtðun unm utz Þg
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we obtain the component C2:
C 2 ¼ Wfv½ð2m nÞunm1 utz þ 2mðn m 1Þunm2 ut uz þ nun1 ð2m nÞDt ðvunm1 uz Þ mDz ðvunm1 ut Þ
Dt Dz ðvunm Þg þ mvunm1 ut Dz ðWÞ vunm Dt Dz ðWÞ þ Dt fv½2munm uz ðn mÞzðu þ munm1 u2z Þ
þ ð2 n mÞtðun unm utz Þg:
Collecting together all the terms containing t again, we get:
C 21 ¼ ð2 n mÞfvun þ t½vt ðun þ munm1 ut uz unm utz Þ þ nvz unm1 u2t þ vtz unm ut g:
The remaining terms in C2 will give:
C 22 ¼ v½2nun ð2 þ n mÞunm utz þ mð2 þ n mÞunm1 ut uz ðn mÞzuvt ½2u ðn mÞzuz ðnvz unm1 ut þ vtz unm Þ
ð36Þ
and the term (n m)zvF, which we can ignore. Thus
C 2 ¼ C 21 þ C 22 ¼ ð2 þ n mÞv½un unm utz þ munm1 ut uz þ vt ½ð2 n mÞtðun þ munm1 ut uz unm utz Þ ðn mÞzu
þ nvz ½ð2 n mÞtunm1 u2t 2unm ut þ ðn mÞzunm1 ut uz þ vtz ½ð2 n mÞtunm ut 2unmþ1
þ ðn mÞzunm uz :
ð37Þ
Thus Eqs. (35) and (37) define the components of a nonlocal conserved vector corresponding to the operator X3.
Local conservation laws
1
we immediately obtain from (35), (37) the local
Remark. The adjoint Eq. (7) has a solution v = const. If we choose v ¼ 2þnm
conserved vector with the components (4),
e 1 ¼ u;
C
e 2 ¼ munm1 ut uz unm utz þ un :
C
Case 1. Eq. (5) is self-adjoint: n = m.
Substitution of v = u in (35) yields
C 1 ¼ uz ½2mzðu þ mu2m1 u2z u2m uzz Þ þ 2ðu2m uz tum Þ þ ð2 2mÞu2
whence
C 1 ¼ ð2 mÞu2m u2z þ ð2 mÞu2 þ Dz ½mzðu2 þ u2m u2z Þ 2tum uz :
Since
(
2tum uz ¼
Dz
2t
m1
ð38Þ
u1m ; if m–1;
Dz ð2t ln uÞ;
ð39Þ
if m ¼ 1;
therefore for n + m = 0, m – 1 the component C1 has the form:
C 1 ¼ ð2 mÞðu2m u2z þ u2 Þ þ Dz ½mzðu2 þ u2m u2z Þ þ Dz
2t
u1m
m1
ð40Þ
and for n + m = 0, m = 1
C 1 ¼ u2 u2z þ u2 þ Dz ½zðu2 þ u2 u2z Þ þ Dz ð2t ln uÞ:
Using (37) and adding
Dt ½mzðu2 þ u2m u2z Þ þ Dt
2t
u1m
m1
ð41Þ
corresponding to the remaining terms in (40) we obtain C2:
C 2 ¼ ð2 2mÞu½um u2m utz þ mu2m1 ut uz þ ut ½2tðum þ mu2m1 ut uz u2m utz Þ þ 2mzu
muz ½2tu2m1 u2t 2u2m ut 2mzu2m1 ut uz þ utz ½2tu2m ut 2u12m 2mzu2m uz þ Dt ½mzðu2 þ u2m u2z Þ
h
i
2t
m
þ Dt
u1m ¼ 2ð2 mÞ mu2m ut uz u12m utz u1m :
m1
1m
Thus for n + m = 0 and m – 1 the density has the form:
C 1 ¼ u2m u2z þ u2
ð42Þ
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and
h
C 2 ¼ 2 mu2m ut uz u12m utz i
m
u1m
1m
ð43Þ
(which also holds for m = 2). Indeed, Dt(C1) + Dz(C2) = 2uF which is equal to zero on solutions of the equation F = 0.
For n = 1 and m = 1 we have the density from (41)
C 1 ¼ u2 u2z þ u2 :
ð44Þ
From (37), after addition of the term Dt ½zðu2 þ u2 u2z Þ þ Dt ð2t ln uÞ corresponding to the remaining terms in (41), we obtain the component C2:
C 2 ¼ ut ½2tðu1 þ u3 ut uz u2 utz Þ þ 2zu uz ½2tu3 u2t 2u2 ut 2zu3 ut uz þ utz ½2tu2 ut 2u1 2zu2 uz þ Dt ½zðu2 þ u2 u2z Þ þ Dt ð2t ln uÞ
whence
C 2 ¼ 2 u2 ut uz u1 utz ln u :
ð45Þ
Thus for n = 1 and m = 1 the conserved vector has the components (44) and (45).
Case 2a. Eq. (5) is quasi-self-adjoint, n + m – 1 and v = u1nm.
According to (35) the density has the following form:
C 1 ¼ ð1 n mÞunm uz ½ðn mÞzðu þ munm1 u2z unm uzz Þ þ ð2 n mÞðunm uz tun Þ þ ð2 þ n mÞu2nm
whence
1
1
ð1 n mÞð4 n 3mÞu2m u2z þ ð2 þ n mÞu2nm Dz ðn mÞð1 n mÞzu2m u2z
2
2
C1 ¼
þ ðn mÞð1 n mÞzu1nm uz ð1 n mÞð2 n mÞtum uz :
ð46Þ
Transformation of two last terms gives the following result:
ðn mÞð1 n mÞzu1nm uz ¼
i
8 h
ðnmÞð1nmÞ
>
zu2nm
>
2nm
> Dz
>
>
>
>
< ðnmÞð1nmÞ
u2nm ;
2nm
if n þ m–2;
>
>
>
>
>
>
> Dz ½2ð1 mÞz ln uÞ þ 2ð1 mÞ ln u
:
if n þ m ¼ 2;
h
i
(
tu1m ; if m–1;
Dz ð1nmÞð2nmÞ
m
1m
ð1 n mÞð2 n mÞtu uz ¼
Dz ½nð1 nÞt ln u;
if m ¼ 1:
It shows that we have four possibilities:
ðiÞ n þ m–2;
m–1;
ðiiÞ n þ m ¼ 2;
m–1;
ðiiiÞ n–1;
m ¼ 1;
ðivÞ n ¼ 1;
m ¼ 1:
When n + m – 2, m – 1 the remaining terms in C1 in (46) are
Dz
1
ðn mÞð1 n mÞ 2nm
ð1 n mÞð2 n mÞ 1m
ðn mÞð1 n mÞzu2m u2z þ Dz
zu
tu
Dz
;
2
2nm
1m
ð47Þ
for n + m = 2, m – 1
Dz ½ð1 mÞzu2m u2z Dz ½2ð1 mÞz ln uÞ;
ð48Þ
and for n – 1, m = 1
Dz
hn
2
i
ðn 1Þzu2 u2z þ Dz ½nzu1n þ Dz ½nð1 nÞt ln u:
ð49Þ
Case (i) n + m – 2, m – 1.
C1 has the form:
C1 ¼
1
ðn mÞð1 n mÞ 2nm
ð1 n mÞð4 n 3mÞu2m u2z þ ð2 þ n mÞu2nm u
2
2nm
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or
C 1 ¼ ð4 n 3mÞ
1
1
ð1 n mÞu2m u2z þ
u2nm :
2
2nm
ð50Þ
Using (37) and (48) we compute the component C2:
C 2 ¼ ð2 þ n mÞu1nm ½un unm utz þ munm1 ut uz þ ð1 n mÞunm ut ½ð2 n mÞtðun þ munm1 ut uz unm utz Þ
ðn mÞzu þ nð1 n mÞunm uz ½ð2 n mÞtunm1 u2t 2unm ut þ ðn mÞzunm1 ut uz þ ð1 n mÞ½ðn þ mÞunm1 ut uz þ unm utz ½ð2 n mÞtunm ut 2unmþ1 þ ðn mÞzunm uz 1
ðn mÞð1 n mÞ 2nm
ð1 n mÞð2 n mÞ 1m
Dt
Dt ðn mÞð1 n mÞzu2m u2z þ Dt
zu
tu
2
2nm
1m
whence
h
C 2 ¼ ð4 n 3mÞ mu2m ut uz u12m utz þ
i
n
u1m :
1m
ð51Þ
Thus for n + m – 1, n + m – 2 and m – 1 it follows from (50) and (51) that the components
C1 ¼
1
1
ð1 n mÞu2m u2z þ
u2nm :
2
2nm
ð52Þ
and
C 2 ¼ mu2m ut uz u12m utz þ
n
u1m
1m
ð53Þ
define a nontrivial conserved vector.
Case (ii) n + m = 2, m – 1.
For these values of m and n the term
1
ð4 n 3mÞð1 n mÞu2m u2z þ ð2 þ n mÞu2nm
2
in (46) becomes equal to ð1 mÞu2m u2z þ 2ð1 mÞ. Therefore we have
1
C 1 ¼ ð1 mÞu2m u2z þ 2ð1 mÞ ln u ¼ 2ð1 mÞ u2m u2z þ ln u :
2
Ignoring the constant 2(1 m) we can choose
1
C 1 ¼ u2m u2z þ ln u
2
ð54Þ
2
as the density of the conservation law in this case. Invoking (48) we have the corresponding component C from (37):
C 2 ¼ 2ð2 mÞu1 ½u2m u22m utz þ mu12m ut uz þ u2 ut ½ð2 2mÞzu ð2 mÞu2 uz ½2u22m ut þ ð2 2mÞzu12m ut uz þ ½2u3 ut uz u2 utz ½2u32m þ ð2 2mÞzu22m uz þ Dt ½ð1 mÞzu2m u2z Dt ½2ð1 mÞz ln uÞ
or
ð2 mÞ 1m
:
C 2 ¼ 2ð1 mÞ mu2m ut uz u12m utz þ
u
1m
Hence, C2 corresponding to (54), has the form:
C 2 ¼ mu2m ut uz u12m utz þ
ð2 mÞ 1m
u :
1m
ð55Þ
Thus, for n + m = 2 and m – 1, the conserved vector has the components (54) and (55).
Case (iii) n – 1, m = 1.
Substituting m = 1 in (50) we obtain that
n
1
C 1 ¼ ð1 nÞ u2m u2z þ
u1n :
2
1n
ð56Þ
Using (49) we calculate C2 from (37):
C 2 ¼ ð1 þ nÞun ½un un1 utz þ un2 ut uz nun1 ut ½ð1 nÞtðun þ un ut uz un1 utz Þ ðn 1Þzu
n2 un1 uz ½ð1 nÞtun2 u2t 2un1 ut þ ðn 1Þzun2 ut uz n½ðn þ 1Þun2 ut uz þ un1 utz ½ð1 nÞtun1 ut 2un
hn
i
þ ðn 1Þzun1 uz þ Dt ðn 1Þzu2 u2z þ Dt ½nzu1n þ Dt ½nð1 nÞt ln u
2
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or
C 2 ¼ 1 þ n þ ð1 nÞðu2 ut uz u1 utz þ n ln uÞ:
whence (ignoring the constant 1 + n)
C 2 ¼ ð1 nÞðu2 ut uz u1 utz þ n ln uÞ:
ð57Þ
Hence, it follows from (56) and (57), that when n – 1, m = 1, the components of the conserved vector are
n
1
C 1 ¼ u2m u2z þ
u1n ;
2
1n
C 2 ¼ u2 ut uz u1 utz þ n ln u:
ð58Þ
Case (iv) m = 1, n = 1.
In (46) we substitute only m = 1 in the terms which can be equal to 0. Then we have
C1 ¼
1
ð1 nÞu2 u2z þ 2 Dz
2
1n
zu2 u2z þ ðn 1Þzu1 uz þ ð1 nÞtu1 uz :
2
After ignoring the trivial part, the constant 2, we notice that all the other terms are multiplied by 1 n. Therefore the density
has the form:
1
C 1 ¼ u2 u2z þ ln u
2
ð59Þ
and the remaining terms are
Dz
1 n 2 2
zu uz Dz ½ð1 nÞz ln u þ Dz ½ð1 nÞt ln u
2
Repeating the steps for calculating C2 for Case (iii), and replacing Dt[nzu1n] by Dt[(1 n)zlnu] we obtain:
C 2 ¼ ð1 nÞðu2 ut uz u1 utz þ ln uÞ:
Hence
C 2 ¼ u2 ut uz u1 utz þ ln u:
ð60Þ
Thus, for m = 1, n = 1, (59) and (60) define a nontrivial conserved vector.
Case 2b Eq. (5) is quasi-self-adjoint, n + m = 1 and v = ln u.
For the second case of quasi-self-adjointness of Eq. (5) we have the following C1 from (35):
C 1 ¼ u1 uz ½ð1 2mÞzðu þ mu2m u2z u12m uzz Þ þ u12m uz tu1m þ ð3 2mÞu ln u
or
C 1 ¼ ð3 2mÞu ln u þ Dz ½ð1 2mÞzu ð1 2mÞu Dz
where
(
tum uz ¼
Dz
1
1m
tu1m
Dz ðt ln uÞ
1 2m 2m 2
1 2m 2m 2
zu uz þ
u uz þ u2m u2z tum uz
2
2
if m–1
if m ¼ 1; n ¼ 0:
Thus, for n + m = 1 and m – 1, the component C1 has the form:
C1 ¼
1
ð3 2mÞu2m u2z þ ð3 2mÞu ln u ð1 2mÞu:
2
ð61Þ
Eq. (5) is itself a conservation law, which has the density equal to
e 1 ¼ u and
C
e 2 ¼ munm1 ut uz unm utz þ un :
C
e 2 of the conserved vector (4) has the form:
For n + m = 1 and m – 1 the component C
e 2 ¼ mu2m ut uz u12m utz þ u1m :
C
ð62Þ
Hence, (61) is the density of a conservation law, which is the linear combination of two conservation laws. We exclude the
e 1 ¼ ð1 2mÞu from C1 in (61):
term ð1 2mÞ C
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1
C 1 ¼ ð3 2mÞ u2m u2z þ u ln u
2
13
ð63Þ
e 2 to the respective C2. Using (37) and also adding the term corresponding for m – 1 to
and add the term ð1 2mÞ C
Dz ½ð1 2mÞzu Dz
1 2m 2m 2
1
zu uz Dz
tu1m ;
2
1m
we finally obtain:
C 2 ¼ ð3 2mÞ ln u½u1m u12m utz þ mu2m ut uz þ u1 ut ½tðu1m þ mu2m ut uz u12m utz Þ ð1 2mÞzu
þ ð1 mÞu1 uz ½tu2m u2t 2u12m ut þ ð1 2mÞzu2m ut uz þ ðu2 ut uz þ u1 utz Þ½tu12m ut 2u22m
1 2m 2m 2
1
þ ð1 2mÞzu12m uz þ Dt ½ð1 2mÞzu Dt
zu uz Dt
tu1m
2
1m
þ ð1 2mÞ½mu2m ut uz u12m utz þ u1m ð64Þ
whence
n
C 2 ¼ ð3 2mÞ ln u½mu2m ut uz u12m utz þ u1m þ mu2m ut uz u12m utz o
m
u1m :
1m
Thus, it follows from (63) and the latter equation, that if n + m = 1, m – 1, the conserved vector has the following
components:
C1 ¼
1 2m 2
u uz þ u ln u;
2
C 2 ¼ ln u½mu2m ut uz u12m utz þ u1m þ mu2m ut uz u12m utz m
u1m :
1m
ð65Þ
Since we assumed here that n – 0, we investigate case m = 1, n = 0 later on.
Remark. The conserved vector in [11], Case (B.5) is a linear combination of two conserved vectors. It should have the
components (65).
7. Computation of conservation laws
n = 0, m – 0 and m– 43
In this case Eq. (5) admits a group of transformations with the following operators [11]:
X a ¼ aðtÞ
o
;
ot
X2 ¼
o
;
oz
e 3 ¼ mz o þ 2u o ;
X
oz
ou
ð66Þ
where a(t) is an arbitrary function.
We have investigated the operator X2 in Section 6.2. Let us have a look at two other operators.
7.1. Operator X a ¼ aðtÞ oto
For this operator n1 = a(t), n2 = g = 0, therefore W = aut. The component (21) has the form:
C 1 ¼ vfaut ½1 þ mum1 uzz mðm þ 1Þum2 u2z þ 2mum1 uz Dz ðaut Þg Dz ðaut ÞDz ðvum Þ
or
C 1 ¼ av½ut þ mum1 ut uzz mðm þ 1Þum2 ut u2z 2mum1 uz utz autz Dz ðvum Þ:
Using (28) when n = 0 we obtain
C 1 ¼ avum utzz autz Dz ðvunm Þ ¼ Dz ðavum utz Þ:
Thus the operator Xa provides only trivial conservation laws.
e 3 ¼ mz o þ 2u o
7.2. Operator X
oz
ou
Nonlocal conservation law
e 3 ¼ X 3 X a when n = 0 and a(t) = (2 m)t. It means that we can find the components of conserved vectors
The operator X
using the formulae (34) and (36) for C 12 , C 22 . They do not include the terms depending on t. Setting n = 0 in (34) and (36) we
obtain the components C1 and C2:
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C 1 ¼ vz ½mzðu þ mum1 u2z um uzz Þ þ ð2 mÞum uz þ ð2 mÞvu;
ð67Þ
C 2 ¼ v½ð2 mÞum utz þ mð2 mÞum1 ut uz þ mzuvt ½2u1m þ mzum uz vtz :
Thus, for n = 0, m – 0 and m– 43, a nonlocal conserved vector is defined by (67).
Local conservation laws
Since n = 0 but m – 0, we can take into consideration only the quasi-self-adjointness of Eq. (5). We have the following
possibilities:
ðiÞ m–2;
Case (i)
m–1; v ¼ u1m ;
ðiiÞ m ¼ 2;
v ¼ u1 ;
ðiiiÞ m ¼ 1; v ¼ ln u:
follows from (52) and (53):
1
1
ð1 mÞu2m u2z þ
u2m ;
2
2m
C 2 ¼ mu2m ut uz u12m utz ;
C1 ¼
Case (ii)
ð68Þ
from (54) and (55):
1
C 1 ¼ u4 u2z þ ln u;
2
C 2 ¼ 2u4 ut uz u3 utz :
Case (iii).
C1 ¼
ð69Þ
When m = 1 and n = 0 we obtain from (63) the following C1:
1 2 2
u uz þ u ln u
2
ð70Þ
Excluding terms with t in (64) and setting m = 1 we have
C 2 ¼ ln u½1 u1 utz þ u2 ut uz þ zut þ ðu2 ut uz þ u1 utz Þ½2 zu1 uz Dt ½zu þ Dt
1 2 2
zu uz ½u2 ut uz u1 utz þ 1
2
whence
C 2 ¼ ln u½u1 utz þ u2 ut uz þ u2 ut uz u1 utz :
ð71Þ
Remark. The conserved vector in [11], Case (B.6) in Table 2, is a linear combination of two conserved vectors. It should have
the components (70) and (71).
8. Computation of conservation laws, n = 0 and m ¼ 43
For these values of m and n Eq. (5) admits a group of transformations with the operators (66) and the additional operator
[11] is
X 4 ¼ z2
o
o
þ 3zu :
oz
ou
ð72Þ
Hence
W ¼ 3zu þ z2 uz ;
Dz ðWÞ ¼ 3u þ 5zuz þ z2 uzz ;
Dt Dz ðWÞ ¼ 3ut þ 5zutz þ z2 utzz :
8.1. Nonlocal conservation law
We obtain C1 for a nonlocal conservation law from (21):
4 7
28 10 2
8 7
4
C 1 ¼ v ð3zu þ z2 uz Þ 1 þ u3 uzz u 3 uz þ u3 uz ð3u þ 5zuz þ z2 uzz Þ þ ð3u þ 5zuz þ z2 uzz ÞDz ðvu3 Þ
3
9
3
or
h
i
28 10 3
4
4
7
7
4
C 1 ¼ v 8u3 uz þ z 3u þ 4u3 uzz þ 4u3 u2z þ z2 uz þ 4u3 uz uzz þ ð3u þ 5zuz þ z2 uzz ÞDz ðvu3 Þ
u 3 uz
9
ð73Þ
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15
and C2 from (22):
8 7
56 10
8
4
7
7
4
C 2 ¼ ð3zu þ z2 uz Þ v u3 utz u 3 ut uz Dt ðvu3 uz Þ Dz ðvu3 ut Þ Dt Dz vu3
3
9
3
3
4 7
4
2
2
þ vu 3 ut ð3u þ 5zuz þ z uzz Þ vu 3 ð3ut þ 5zutz þ z utzz Þ:
3
Setting in Eq. (5) m ¼ 43 ; n ¼ 0 we obtain:
4
u3 utzz þ
8 7
4 7
28 10
u 3 uz utz þ u3 ut uzz u 3 ut u2z ¼ ut :
3
3
9
Therefore, for the following terms with z2v, we have
8 7
4 7
56 10
28 10 2
4
u3 utzz þ u3 uz utz þ u3 ut uzz u 3 ut u2z ¼ ut u 3 ut uz :
3
3
9
9
Hence
28 10
8
4
4
4
7
7
7
4
C 2 ¼ v u3 ut þ z 3u3 utz 12u3 ut uz z2 ut þ
u 3 ut u2z Dt ðvu3 uz Þ Dz ðvu3 ut Þ Dt Dz vu3 :
9
3
3
ð74Þ
Thus, for m ¼ 43 ; n ¼ 0, the nonlocal conserved vector has the components (73) and (74).
8.2. Local conservation law
1
Since n – m and n + m – 1, we have only one possibility: we can use the quasi-self-adjointness of Eq. (5) when v ¼ u3 :
Invoking (73) we get the following C1:
h 2
i
28 11 3
5 8
5
5
8
1
8
C 1 ¼ 8u3 uz þ z 3u3 þ 4u3 uzz þ 4u3 u2z þ z2 u3 uz þ 4u3 uz uzz u 3 uz u3 uz ð3u þ 5zuz þ z2 uzz Þ
9
3
or
3 2 3
7
5
2
8
C 1 ¼ Dz 4zu3 uz u3 þ z2 u3 þ z2 u3 u2z :
2
2
6
Thus the local conservation law is trivial.
8.3. Remark
In [11] the conserved vector with the components
e1
C1 ¼ zC
e 2 þ u3 ut
C2 ¼ zC
4
e 1, C
e 2 , defined by (4), under the action of the operator X4.
is obtained from the conserved vector with the components C
The operator is admitted by Eq. (5) when n = 0 and m ¼ 43. However, for n = 0, the equation and conservation law have the
form
1m u
¼ 0;
Dt u D2z
1m
Dt ½u D2z ðln uÞ
¼ 0;
if m–1;
if m ¼ 1:
Therefore it is evident that for n = 0 it is possible to multiply the conserved vector by any function f(z). The new conserved
vector will have the following components:
u1m
;
1m
2
m1
ut uz um utz Þ þ f 0 ðzÞum ut
C ¼ f ðzÞðmu
C 1 ¼ f ðzÞu f 00 ðzÞ
and
C 1 ¼ f ðzÞu f 0 ðzÞ ln u;
C 2 ¼ f ðzÞðu2 ut uz u1 utz Þ þ f 0 ðzÞu1 ut
for m – 1 and m = 1, respectively.
I do not investigate the case n = 0, m = 0 because Eq. (5) becomes linear.
Please cite this article in press as: Khamitova R, Symmetries and nonlocal conservation laws of the general magma equation, Commun Nonlinear Sci Numer Simulat (2008), doi:10.1016/j.cnsns.2008.08.009
ARTICLE IN PRESS
16
R. Khamitova / Commun Nonlinear Sci Numer Simulat xxx (2008) xxx–xxx
9. Conclusion
The direct method for calculating conserved vectors of the general magma equation is complicated. On the other hand,
the use of the theorem on nonlocal conservation laws [1] and self-evident symmetries of the general magma equation (5)
gives a simple regular algorithm for computing conservation laws.
References
[1] Ibragimov NH. A new conservation theorem. J Math Anal Appl 2007;333(1):311–28.
[2] Ibragimov NH. Quasi-self-adjoint differential equations. Preprint in Archives of ALGA, vol. 4. BTH, Karlskrona, Sweden: Alga Publications; 2007. p. 55–
60, ISSN: 1652-4934.
[3] Takahashi D, Sachs JR, Satsuma J. Properties of the magma and modified magma equations. J Phys Soc Jpn 1990;59(6):1941–53.
[4] Scott DR, Stevenson DJ. Magma solutions. Geophys Res Lett 1984;11:1161–4.
[5] Dullien FAL. Porous media fluid transport and pure structure. New York: Academic Press; 1979.
[6] Atherton RW, Homsy GM. On the existence and formulation of variational principles for nonlinear differential equations. Studies Appl Math
1975;LIV(1):31.
[7] Khamitova R. Self-adjointness and quasi-self-adjointness of an equation modelling melt migration through the Earth’s mantle. Nonlocal conservation
laws. Preprint in Archives of ALGA, vol. 5. BTH, Karlskrona, Sweden: Alga Publications; 2008. p. 143–58, ISSN: 1652-4934.
[8] Khamitova R. In: Tenreiro Machado JA, Silva MF, Barbosa RS, Figueiredo LB, editors. Proceedings of the 2nd conference on nonlinear science and
complexity, July 28–31, 2008, Porto, Portugal, ISBN: 978-972-8688-56-1.
[9] Barcilon V, Ritcher FM. Nonlinear waves in compacting media. J Fluid Mech 1986;164:429–48.
[10] Harris SE. Conservation laws for a nonlinear wave equation. Nonlinearity 1996;9:187–208.
[11] Maluleke GH, Mason DP. Derivation of conservation laws for a nonlinear wave equation modelling melt migration using Lie point symmetry
generators. Commun Nonlinear Sci Numer Simulat 2007;12(4):423–33.
Please cite this article in press as: Khamitova R, Symmetries and nonlocal conservation laws of the general magma equation, Commun Nonlinear Sci Numer Simulat (2008), doi:10.1016/j.cnsns.2008.08.009
Acta Wexionensia
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165. Katarina H. Thorén, 2008 “Activation Policy in Action”: A Street-Level Study of
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166. Lennart Karlsson, 2009. Arbetarrörelsen, Folkets Hus och offentligheten i Bromölla
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168. Martin Estvall, 2009. Sjöfart på stormigt hav – Sjömannen och Svensk Sjöfarts Tidning inför den nazistiska utmaningen 1932-1945 (doktorsavhandling). ISBN: 978-917636-647-9.
169. Cecilia Axelsson, 2009. En Meningsfull Historia? Didaktiska perspektiv på historieförmedlande museiutställningar om migration och kulturmöten (doktorsavhandling).
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170. Raisa Khamitova, 2009. Symmetries and conservation laws (doktorsavhandling).
ISBN: 978-91-7636-650-9.
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