Instability of local deformations of an
elastic filament
with
S. Lafortune & S. Madrid-Jaramillo
Department of Mathematics
University of Arizona
S. Lafortune & J.L., Physica D 182, 103-124 (2003)
S. Lafortune & J.L., submitted to SIMA
S. Lafortune, J.L. & S. Madrid-Jaramillo, submitted to Chaos
University of Arizona
Tucson AZ
Outline

Motivations
–
–
–

Evans function methods
–
–
–
–
–

[S. Lafortune & J.L., Physica D 182, 103-124 (2003)]
Linearization about pulse solutions
Behavior of the Evans function at the origin
Behavior of the Evans function at infinity
Instability criterion
Numerical evaluation of the Evans function
[S. Lafortune, J.L. & S. Madrid-Jaramillo, submitted to Chaos]
Hamiltonian formalism
–
–

Dynamics of an elastic filament
The coupled Klein-Gordon equations
Special solutions and numerical simulations of pulse solutions
[S. Lafortune & J.L, submitted to SIMA]
Hamiltonian form of the coupled Klein-Gordon equations
Spectral stability criterion
Conclusions
Dynamic of an elastic filament

Consider an elastic
filament kept under tension
and subject to constant
twist

There is a critical value of
the applied twist above
which the filament
undergoes a writhing
bifurcation
Description of near-threshold dynamics
“Reconstruction”
of elastic filament
Elastic
properties and
parametrization of
elastic filament
The Kirchhoff
equations
Near-threshold
dynamics
The coupled
Klein-Gordon
equations
Envelope equations
The coupled Klein-Gordon equations


They describe the near-threshold dynamics of an elastic
filament subject to sufficiently high constant twist
Dimensionless form of the equations
2 A 2 2 A
B
2
 c0 2   A | A | A  A
2
t
x
x
2B 2B
 | A |2
 2 
2
t
x
x
A: complex envelope of helical mode
B: axial twist
A. Goriely & M. Tabor, Nonlinear dynamics of filaments II: Nonlinear analysis, Physica D
105, 45-61 (1997).
Some special solutions

Traveling holes

Traveling fronts

Periodic solutions

Traveling pulses
J.L. & A. Goriely, Pulses, fronts and oscillations of an elastic rod, Physica D 132, 373-391
(1999).
Traveling pulse solutions

Analytic expression
A  a ( ) exp( it ), B  b( ),   x  ct
 c


a ( )   sech (  ) exp  i 2

2
 c  c0 
b( ) 
2
 (1  c )
2
tanh(  )
2( c 2  1)
  2 2 2  (c 2  c02 )   2c02 
c (c  c0 )
2
 
2
 (c 2  c02 )   2c02
(c 2  c02 ) 2
 0
J.L. & A. Goriely, Pulses, fronts and oscillations of an elastic rod, Physica D 132, 373-391
(1999).
Numerical simulations of pulse solutions

Numerical simulation
–
 0
  864
c02  2.4
c  0.6124
L 1
–
Non-reflecting boundary
conditions
–
Compact finite differences
+ Runge-Kutta in time
J.L. & A. Goriely, Pulses, fronts and oscillations of an elastic rod, Physica D 132, 373-391
(1999).
Question

Numerical simulations indicate that some pulses are stable and
some are not

Can one find a criterion which guarantees the instability of the
pulse solutions? If so, this condition should depend on the
speed of propagation c of each pulse

Below, we present two complementary ways - Evans function
methods and Hamiltonian techniques - of answering this
question
Outline

Motivations
–
–
–

Evans function methods
–
–
–
–
–

[S. Lafortune & J.L., Physica D 182, 103-124 (2003)]
Linearization about pulse solutions
Behavior of the Evans function at the origin
Behavior of the Evans function at infinity
Instability criterion
Numerical evaluation of the Evans function
[S. Lafortune, J.L. & S. Madrid-Jaramillo, submitted to Chaos]
Hamiltonian formalism
–
–

Dynamics of an elastic filament
The coupled Klein-Gordon equations
Special solutions and numerical simulations of pulse solutions
[S. Lafortune & J.L, submitted to SIMA]
Hamiltonian form of the coupled Klein-Gordon equations
Spectral stability criterion
Conclusions
Linear stability analysis of
the pulse solutions

Coupled Klein-Gordon equations
2 A 2 2 A
B
2
 c0 2   A | A | A  A
2
t
x
x
2B 2B
 | A |2
 2 
2
t
x
x

Pulse solutions
A  a ( ) exp( it )
B  b( )
  x  ct
Linearization about the pulse solutions

Write
A  a ( )  u( , t )  exp( i t )
B  b( )  w( , t )
where a and b correspond to
a pulse solution

Y
LY
t
Obtain a linearized system of the form
where Y  u, u , w, u / t , u / t , w / t T
is a six-dimensional vector and L is a differential operator in 

Define an eigenfunction of L with eigenvalue l as a solution Y
of L Y = l Y such that (u,ū,w)(H1×H1×L)(C1×C1×C1)
The equation L Y = l Y can be written
dX
X 
 A ( , l ) X
as a six-dimensional system of the form
d
'
The Evans function




An eigenfunction exists if the two vector spaces of solutions of
X’ = A(, l) X that are bounded at + and – intersect nontrivially
Typically, this only happens for particular values of l
The Evans function E(l) is an analytic function of l, which is
real-valued for l real and which vanishes on the point
spectrum of L
It can be viewed as the Wronskian, calculated for instance at
 = 0, of linearly independent solutions of X’ = A(, l) X that
converge at +
Solution
Solution
and linearly
bounded at +
bounded at –
independent solutions
of X’ = A(, l) X
that converge at –
The Evans function

References:
–
–
–
–
–
–
–
–
J.W. Evans, Nerve axon equations. IV. The stable and unstable impulse, Indiana Univ.
Math. J. 24, 1169-1190 (1975).
C.K.R.T. Jones, Stability of the travelling wave solution to the FitzHugh-Nagumo
equation, Trans. AMS 286, 431-469 (1984).
E. Yanagida, Stability of fast traveling pulse solutions to the FitzHugh-Nagumo
equations, J. Math. Biol. 22, 81-104 (1985).
J. Alexander, R. Gardner & C. Jones, A topological invariant arising in the stability
analysis of travelling waves, J. Reine Angew. Math. 410, 167-212 (1990).
R. Pego & M. Weinstein, Eigenvalues, and instabilities of solitary waves, Phil. Trans. R.
Soc. London A 340, 47-94 (1992).
R.A. Gardner & K. Zumbrun, The Gap Lemma and geometric criteria for instability of
viscous shock profiles, Commun. Pure Appl. Math. LI, 0797-0855 (1998).
T. Kapitula & B. Sandstede, Stability of bright solitary-wave solutions to perturbed
nonlinear Schrödinger equations, Physica D 124, 58-103 (1998).
B. Sandstede, Stability of travelling waves, in Handbook of Dynamical Systems II:
Towards Applications, pp. 983-1055, B. Fielder Ed., Elsevier, 2002.
The Evans function

As   ±, A(, l)  A0(l), where the matrix A0(l) has
constant coefficients


If the “deviator”  | A ( , l )  A0 (l ) | d converges and if the

asymptotic matrix A0(l) is diagonalizable, then for each (wi,ni)
such that A0(l) wi = ni wi , there is a solution Fi,l to

X’ = A(, l) X such that lim Fi ( , l ) exp( n i )  wi
 
Moreover, if the convergence of the deviator is uniform in l
on compact subsets of the complex plane, one can find
solutions Fi,l which are analytic in l in some subset of the
complex plane
E. Coddington & N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill,
New-York (1955).
The Evans function


When l > 0, the asymptotic matrix A0(l) has 3 positive and 3
negative eigenvalues
We define the Evans function E(l) by
E(l )  F1 (0, l ), F 2 (0, l ), F3 (0, l ), F 4 (0, l ), F5 (0, l ), F6 (0, l )
where
lim F i ( , l ) exp( νi )  wi , vi  0, i  1,2,3
 
lim F i ( , l ) exp( νi )  wi , vi > 0, i  4,5,6
 

We look for a criterion, involving the speed of the traveling
pulse, which guarantees that E(l) vanishes on the real axis. This
is done by comparing the sign of E(l) near the origin with its
sign for large positive values of the spectral parameter l
Behavior of E(l) near the origin



The Evans function, as well as its first four derivatives, vanish
at l = 0
The fifth derivative has to be computed by finding expansions
of the Fi,l in powers of l
Because of the symmetry B  B + constant of the original
nonlinear Klein-Gordon equations, two of the Fi,l are
bounded at l = 0, and singular expansions of the form
F i ( , l )  exp(n i (l ) ) F i( 0) ( )  l Fi(1) ( )  

are needed in this particular case
The fifth derivative of the Evans function at l = 0 is given by
E ( 5)
128 c02  2 3c02 (3c 2  2c02 )  c 4 (c02  2) 
 5!
3
c6 (c 2  c02 )4
Behavior of E(l) at infinity





The change of variable z = l , turns the system X’ = A(, l) X
into X’ = Ã(z, l) X
As l  +, Ã(z, l) converges uniformly toward a matrix Ã0,
which has constant coefficients
Because X’ = Ã(z, l) X has an exponential dichotomy for l
large, its solutions are uniformly close to solutions of
X’ = Ã0 X for l large enough
The sign of E(l) for l large is therefore the same as the sign of
~
the Evans function E associated with X’ = Ã0 X, provided the
basis vectors are arranged in the same order
One finds
512c02 4
~
E
 0 if   0
2
2
2 3
 (c  1)( c  c0 )
Instability criterion



For l positive and small,


128 c02  2 3c02 (3c 2  2c02 )  c 4 (c02  2) 5
6
E (l ) 
l

O
l
3
c 6 (c 2  c02 ) 4
For  > 0, the asymptotic state of the pulse is unstable to plane
waves perturbations, so we assume  < 0. Then, E(l) is
negative for l positive and large
Therefore, E(l) vanishes for a real positive value of l if
T  3c02 (3c 2  2c02 )  c 4 (c02  2) > 0 , which can be rewritten as
 
2
c02 9  33  24c0
2
2
K

c

c
0 1
2
2
c0  2

In the numerical simulation shown before, T = – 27.078 < 0
Numerical evaluation of the Evans function


One can numerically evaluate the Evans function for l on the
real axis and see how E(l) changes as c2 gets close to K
Below,   216; c0  0.8; cth  K  0.76277007
S. Lafortune, J.L. & S. Madrid-Jaramillo, Instability of local deformations of an elastic rod:
numerical evaluation of the Evans function, submitted to Chaos.
Numerical evaluation of the Evans function


Contour integration in the
complex plane can also be
used to detect the zeros of
the Evans function in a
region near the origin
Here, the winding number is
equal to 6 for
  216; c0  0.8
c  0.76274  K
 0.01  Re( l ), Im( l )  0.01
S. Lafortune, J.L. & S. Madrid-Jaramillo, Instability of local deformations of an elastic rod:
numerical evaluation of the Evans function, submitted to Chaos.
Outline

Motivations
–
–
–

Evans function methods
–
–
–
–
–

[S. Lafortune & J.L., Physica D 182, 103-124 (2003)]
Linearization about pulse solutions
Behavior of the Evans function at the origin
Behavior of the Evans function at infinity
Instability criterion
Numerical evaluation of the Evans function
[S. Lafortune, J.L. & S. Madrid-Jaramillo, submitted to Chaos]
Hamiltonian formalism
–
–

Dynamics of an elastic filament
The coupled Klein-Gordon equations
Special solutions and numerical simulations of pulse solutions
[S. Lafortune & J.L, submitted to SIMA]
Hamiltonian form of the coupled Klein-Gordon equations
Spectral stability criterion
Conclusions
Hamiltonian formalism

The coupled nonlinear Klein-Gordon equations form a
Hamiltonian system
A  P  iQ
v
 J E ( v )
t
At  R  iS
Bt  U
v  ( R , S , U , P , Q , B )T

E ( v )   h dx

1
h  c02 ( Px2  Qx2 )   ( P 2  Q 2 )  ( P 2  Q 2 ) 2  ( P 2  Q 2 ) Bx
2
1 2 1 2
2
2
 R  S  Bx  U
2
2
1 / 2 0 0 


 03   

J  
   0 1/ 2 0
  03 
 0
0 1 

J.L. & A. Goriely, Pulses, fronts and oscillations of an elastic rod, Physica D 132, 373-391 (1999)
Hamiltonian formalism

If v0 corresponds to a pulse solution, then the linearization of
the Hamiltonian system about v0 is of the form
w
 J H c, w
t

where Hc, is self-adjoint
The idea is to use the fact that the spectrum of Hc, is relatively
simpler to analyze, to get information on the spectrum of
J Hc,
M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of
symmetry, I, J. Functional Analysis 74, 160-197 (1987)
M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of
symmetry, II, J. Functional Analysis 94, 308-348 (1990)
Hamiltonian formalism



Assume that the continuous spectrum of Hc, is positive and
bounded away from the origin, and that the Hilbert space in
which perturbations live is the direct sum of the kernel, positive
and negative subspaces of Hc,.
Let
d(c,) = E(v0) – cQ1(v0) – Q2(v0),
where v0 corresponds to the pulse solution, Q1(v0) is the
conserved quantity associated with translational invariance and
Q2(v0) is the conserved quantity associated with gauge
invariance.
Let d″(c,) be non-singular, n(Hc,) be the dimension of the
negative subspace of Hc,, and p(d″) be the number of positive
eigenvalues of the Hessian of d.
Hamiltonian formalism

Theorem (M. Grillakis, J. Shatah and W. Strauss)
–
If n(Hc,) – p(d″) is odd, then J Hc, has at least one pair of
real non-zero eigenvalues.
–
If n(Hc,) = p(d″), then pulse solutions are (orbitally) stable.
M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of
symmetry, II, J. Functional Analysis 94, 308-348 (1990)
Hamiltonian formalism

Here, one can prove that n(Hc,) = 1

But the continuous spectrum of Hc, touches the origin, so the
above theorem is not directly applicable

However, one has the following result:
Let d″(c,) be non-singular. Then, pulses are spectrally stable if and only
if
det( d )  0
S. Lafortune & J.L., Spectral stability of local deformations of an elastic rod: Hamiltonian
formalism, submitted to SIMA
Hamiltonian formalism

This condition reads
 12c 2c04 (c02  1)( 3c02  2c 2  c 4 ) 4
 18c02 (c 4  3c 2c02  c 2  c02 )( c 2  c02 ) 2  2
 3(c 2  1)  2 (c 4c02  2c 4  9c 2c02  6c04 )( c 2  c02 ) 2  0

In the case  = 0, this is consistent with the instability criterion
obtained by means of Evans function techniques
T  3c02 (3c 2  2c02 )  c 4 (c02  2) > 0
Summary: spectrum of linearized operator
Conclusions


Instability of pulse solutions to the coupled nonlinear KleinGordon equations
–
Numerical simulations show that both stable and unstable pulses exist
–
This analysis indicates that pulses propagating at the speed c are
spectrally stable if and only if c2 is less than K  c02
Evans function techniques
–
The Evans function as well as its first 4 derivatives vanish at the origin
–
The spaces of solutions which are bounded at plus infinity and at minus
infinity are 3-dimensional
Conclusions

Evans function techniques
–
Regular and singular perturbation expansions of the solutions in powers
of l are needed to calculate the fifth derivative of the Evans function at
the origin
–
The sign of the Evans function for large and positive values of the
spectral parameter is found by calculating the Evans function of a
system with constant coefficients (obtained after a suitable change of
variable)
–
Analytical results can be complemented by a numerical evaluation of
the Evans function
Conclusions

Hamiltonian formalism
–
Because the coupled Klein-Gordon form a Hamiltonian system, it is
possible to obtain a necessary and sufficient condition for spectral
stability
–
Classical Hamiltonian methods have to be adapted to this case because
the continuous spectrum of the modified Hamiltonian touches the
origin
–
It may be possible to obtain strong stability results provided the
perturbations live in a different space
Conclusions

Hamiltonian formalism
–
Because the coupled Klein-Gordon form a Hamiltonian system, it is
possible to obtain a necessary and sufficient condition for spectral
stability
–
Classical Hamiltonian methods have to be adapted to this case because
the continuous spectrum of the modified Hamiltonian touches the
origin
–
It may be possible to obtain strong stability results provided the
perturbations live in a different space
Symmetries

The coupled Klein-Gordon
equations are
–
–
–
Invariant under space translations
Gauge invariant
Invariant under B  B + constant
2 A 2 2 A
B
2

c


A

|
A
|
A

A
0
t 2
x 2
x
2B 2B
 | A |2


t 2 x 2
x
A  a ( ) exp( it ), B  b( ),   x  ct

Two-parameter family pulse of
solutions
–
–
The speed c is associated with the
space translation invariance
The frequency  is associated
with the gauge invariance
 c


a ( )   sech (  ) exp  i 2

2
c

c
0


b( ) 
2
 (1  c 2 )
tanh(  )
2( c 2  1)
  2 2 2  (c 2  c02 )   2c02 
c (c  c0 )
2
 
2
 (c 2  c02 )   2c02
(c 2  c02 ) 2
Traveling pulse solutions

Analytic expression
A  a ( ) exp( it ), B  b( ),   x  ct
 c


a ( )   sech (  ) exp  i 2

2
 c  c0 
b( ) 
2
 (1  c )
2
tanh(  )
2( c 2  1)
  2 2 2  (c 2  c02 )   2c02 
c (c  c0 )
2
 
2
 (c 2  c02 )   2c02
(c 2  c02 ) 2
J.L. & A. Goriely,
Pulses, fronts and oscillations of an elastic rod,
Physica D 132, 373-391 (1999).
 0