Advanced Studies in Theoretical Physics
Vol. 11, 2017, no. 3, 105 - 114
HIKARI Ltd, www.m-hikari.com
https://doi.org/10.12988/astp.2017.61243
A Solution of the One-Dimensional
Nonlinear Logarithmic Schrödinger Equation
F. Fonseca
Universidad Nacional de Colombia
Grupo de Ciencia de Materiales y Superficies
Departamento de Fı́sica
Bogotá, Colombia
c 2016 F. Fonseca. This article is distributed under the Creative Commons
Copyright Attribution License, which permits unrestrikd use, distribution, and reproduction in any
medium, provided the original work is properly cited.
Abstract
In this paper we find solutions of the Nonlinear Logarithmic Schrödinger
equation. We use a solitary wave and field transformations that let us
to find analytical solutions in terms of hyperbolic and Jacobi elliptic
functions. We present the spatio-temporal evolution of the wave function, and several structure profiles as the nonlinear coupling constants
change.
Keywords: Nonlinear Logarithmic Schrödinger equation, hyperbolic functions, Jacobi elliptic functions
1
Introduction
The Schrödinger equation (SEq) is one of the fundamental tools in the physical
description of quantum world. The phenomenology offered by this equation
is greatly enriched by the nonlinear terms added to it. We can find nonlinear
Terms like power laws, e.g. cubic or quintic, functions as sin cos or sinh of the
wave function. Recently, in the work reported in [1], [2], terms proportional to
the logarithm of arithmetic combinations of the wave function are incorporated
to SEq. Those logarithmic terms are relevant in research in quantum field
theory [3]-[4] and critical phenomena [5]-[6].
106
F. Fonseca
This work presents the solution of the logarithmic Schrödinger equation. In
section (2), it is shown the coordinate and field transformations that let us to
find the real and complex equations. In sections (3) and (4), in order to find
solutions we apply the hyperbolic and Jacobi elliptic functions, respectively.
Finally, we present conclusions in section (5).
2
One-dimensional Nonlinear Logarithmic
Schrödinger equation
We start from the one-dimensional logarithmic Schrödinger equation, [1]:
h̄2 ∂ 2 ψ
ψ
∂ψ
2
=
−
ih̄(κ
ln
|ψ|
+
Γ
ln
)ψ
ih̄
∂t
2m dx2
ψ̄
(1)
Firstly, we start doing the following coordinate transformation:
u = x − ct + u0 ;
h̄ = 1;
2m = 1
(2)
The derivatives are:
Figure 1: Spatio-temporal evolution of the traveling wave solution 1 v(x − ct).
107
Nonlinear logarithmic Schrödinger equation
Figure 2: The traveling wave solution 1, v(x−ct). The parameters are a = 0.4,
c = 1 κ = 2.1 and Γ = 2.1.
d
∂
=
;
∂x
du
∂
d
= −c ;
∂t
du
∂2
d2
=
∂x2
du2
(3)
Replacing in eq. (1)
dψ
d2 ψ
ψ
=
− i(κ ln |ψ|2 + Γ ln )ψ
2
du
du
ψ̄
Also, we define the next field transformation
−ci
ψ = exp (v)
(4)
(5)
Then
dψ
dψ dv
dv d2 ψ
dv 2
d2 v
=
= exp (v) ;
=
exp
(v)(
)
+
exp
(v)
du
dv du
du du2
du
du2
And replacing in eq. (4)
−ci exp (v)
dv
dv
d2 v
Ψ
= exp (v)( )2 + exp (v) 2 − i(κ ln |Ψ|2 + Γ ln )Ψ
du
du
du
Ψ̄
Assuming v = vr + ivc
(6)
(7)
108
F. Fonseca
Ψ
exp (v)
= exp (2ivc )
=
exp (v̄)
Ψ̄
(8)
dv
dv
d2 v
= exp (v)( )2 + exp (v) 2 − i(κ2vr + Γ2ivc ) exp (v)
du
du
du
(9)
|Ψ|2 = ΨΨ̄ = exp (v) exp (v̄) = exp (2vr );
−ci exp (v)
Figure 3: The traveling wave solution 1, v(x − ct). The parameter κ goes from
2.1, 3.1 to 4.1. The parameters are a = 0.4, c = 1 and Γ = 2.1.
The real component equation
dvr 2
dvc 2 d2 vr
dvc
=(
) +(
) +
+ Γ2vc
du
du
du
du2
The complex component equation
c
−c
3
d2 v c
dvr
= + 2 − κ2vr
du
du
Solution 1
Now, we suppose the real and complex solution are:
(10)
(11)
109
Nonlinear logarithmic Schrödinger equation
Figure 4: The traveling wave solution 1, v(x − ct). The parameter Γ goes from
2.1, 4.1 to 6.1. The parameters are a = 0.4, c = 1 and κ = 2.1.
vr = A sinh (au);
vc = iA cosh (au);
dvr
d2 vr
= aA cosh (au);
= a2 A sinh (au)
du
du2
(12)
d2 vc
dvc
= iaA sinh (au);
= ia2 A cosh (au)
du
du2
(13)
The complex component equation is
−caA cosh (au) = ia2 A cosh (au) − κ2A sinh (au) → tanh (au) =
ca + ia2
(14)
2κ
The real component equation is
icaA sinh (au) = (aA)2 cosh2 (au) − (aA)2 sinh2 (au)
+a2 A sinh (au) + 2iΓA cosh (au)
icaA sinh (au) = (aA)2 + a2 A sinh (au) + 2iΓA cosh (au)
(15)
(16)
110
F. Fonseca
and using eq. (14)
cosh (au) =
(a2 A)
2
4
( (ca)2κ−a − 2Γ)
; sinh (au) =
(a2 A(ca + a2 ))
(ca)2 − a4 − 4κΓ
(17)
Using cosh2 (au) − sinh2 (au) = 1, we get:
A=
v
u ca 2
u( )
±t κ
2
2
−a
+ ( c 2κ
−
2
2
2
1 − c a4κ+a
2
2Γ 2
)
a2
(18)
Therefore
v = vr + ivc = A sinh (au) + iA cosh (au)
(19)
ψ = exp (A sinh (au) + iA cosh (au)); u = x − ct + u0
(20)
The solution is
Where A is given by the negative root of eq. (18).
4
Solution 2
Also, we use the Jacobi elliptic functions. They satisfy the next relations:
sn2 (t, k) + cn2 (t, k) = 1, k 2 sn2 (t, k) + dn2 (t, k) = 1
(21)
dn2 (t, k) − k 2 cn2 (t, k) = k 02 , k 02 sn2 (t, k) + cn2 (t, k) = dn2 (t, k)
(22)
k0 =
√
1 − k2
(23)
Then, we suppose the next solutions
vr = Asn(au, k);
dvr
= aAcn(au, k)dn(au, k);
du
d2 v r
= Aa2 (2k 2 sn3 (au, k) − (1 + k 2 )sn(au, k))
du2
(24)
111
Nonlinear logarithmic Schrödinger equation
vc = Acn(au, k);
dvc
= −aAsn(au, k)dn(au, k);
du
(25)
d2 vc
= Aa2 (−2k 2 cn3 (au, k) − (1 − 2k 2 )cn(au, k))
du2
Then, replacing in eqs. (10) and (11)
−acAsn(au, k)dn(au, k) = (aA)2 − (aA)2 k 2 sn2 (au, k) + 2ΓAcn(au, k)(26)
−2Aa2 k 2 cn(au, k) + 2Aa2 k 2 cn(au, k)sn2 (au, k) − Aa2 (1 − 2k 2 )cn(au, k)
The complex component equation
−caAcn(au, k)dn(au, k) = −2Aa2 k 2 cn(au, k)
(27)
2 2
2
2
2
+2Aa k cn(au, k)sn (au, k) − Aa (1 − 2k )cn(au, k) − κ2Asn(au, k)
Equating the left hand sides of eqs. (26) and (27), we get:
−(aA)2 k 2 sn2 (au, k) + κ2Asn(au, k) + (aA)2 + 2ΓAcn(au, k) = 0
(28)
Then, we get:
−(aA)2 k 2 sn2 (au, k) + κ2Asn(au, k) = 0 → sn(au, k) =
(aA)2 + 2ΓAcn(au, k) = 0 → cn(au, k) = −
Aa2
2Γ
2κ
Aa2 k 2
(29)
(30)
So, applying eq. (21)
(
a2 2 4
2κ
) A − A 2 + ( 2 2 )2 = 0
2Γ
ak
(31)
Defining
c1 = (
c3 =
v
u
u
t
1
−
c1
√
a2 2
2κ
) , c2 = ( 2 2 )2
2Γ
ak
1 − 4c1 c2
, c4 =
c1
v
u
u
t
1
+
c1
(32)
√
1 − 4c1 c2
c1
(33)
112
F. Fonseca
Figure 5: The traveling wave solution 2, v(x − ct). The parameter k is taken
as (0, 0.4, 0.7, 0.9). The parameters are a = 1.4, Γ = 2.1 and changes as
κ = 2.5, 4.5, 6.5.
The solutions are:
c3
c4
c4
c3
A1 = − √ , A2 = √ , A3 = − √ , A4 = √
2
2
2
2
(34)
v = vr + ivc = Asn(au, k) + iAcn(au, k)
(35)
ψ = exp (Asn(au, k) + iAcn(au, k)); u = x − ct + u0
(36)
So,
The solution is
Where A is given by eqs. (34).
Nonlinear logarithmic Schrödinger equation
5
113
Conclusions
Finally, we solve the nonlinear logarithmic Schrödinger equation. Also, figures
(1-5), we get the main overall behavior of the nonlinear parameters, κ and Γ,
given in [1]. In addition, the nonlinear part of the equation is solved using two
types of transformations that allow us to find hyperbolic solutions. Also, we
find four sets of solitary wave solutions using elliptic Jacobi functions. They
are:
ψ1 = exp (A sinh (au) + iA cosh (au)); u = x − ct + u0
ψ2 = exp (Asn(au, k) + iAcn(au, k)); u = x − ct + u0
(37)
The extension of the results to higher dimensions is straightforward. As a future work we can extend the method to explore other kind of nonlinear terms.
Acknowledgements. This research was supported by Universidad Nacional
de Colombia in Hermes project (32501).
References
[1] T. Yamano, Localized traveling wave solution for a logarithmic nonlinear
Schrödinger equation, Wave Motion, 67 (2016), 116-120.
https://doi.org/10.1016/j.wavemoti.2016.07.005
[2] H. Hossieni, Solitary Solution of Non-linear Schrdinger Field with a Logarithmic Non-Linearity for Free Particle in (1+1) Dimension, International
Journal of Basic & Applied Sciences IJBAS-IJENS, 13 (2013), no. 02, 1820.
[3] K. G. Zloshchastiev, Logarithmic nonlinearity in theories of quantum
gravity: Origin of time and observational consequences, Grav. Cosmol.,
16 (2010), 288-297. https://doi.org/10.1134/s0202289310040067
[4] K. G. Zloshchastiev, Spontaneous symmetry breaking and mass generation as built-in phenomena in logarithmic nonlinear quantum theory, Acta
Phys. Polon. B, 42 (2011), 261-292.
https://doi.org/10.5506/aphyspolb.42.261
[5] K. G. Zloshchastiev, Vacuum Cherenkov effect in logarithmic nonlinear
quantum theory, Phys. Lett. A, 375 (2011), 2305-2308.
https://doi.org/10.1016/j.physleta.2011.05.012
114
F. Fonseca
[6] A. V. Avdeenkov and K.G. Zloshchastiev, Quantum Bose liquids with logarithmic nonlinearity: Self-sustainability and emergence of spatial extent,
J. Phys. B: At. Mol. Opt. Phys., 44 (2011), 195303.
https://doi.org/10.1088/0953-4075/44/19/195303
Received: December 23, 2016; Published: January 16, 2017