ABSTRACT (original book): Throughout the long history of the nonlinear Schrödinger equation (NLSE), many exact analytical solutions have been found and they continue to grow as new solutions are being sought and discovered. This book aims to organize the solutions by classifying and grouping them based on aspects and symmetries they possess. The authors present a systematic derivation of many solutions and even include new derivations. This expanded second edition contains new solutions published or derived since the first edition. Noting the increasing interest in and applications of the fractional nonlinear Schrödinger equation, a new chapter devoted to this topic has been added. Each chapter now also features an introductory section documenting the history, background, and physical systems described by the equations at hand. CITATION (original book): Al Khawaja, Usama and Al Sakkaf, Laila (2024), Handbook of Exact Solutions to the Nonlinear Schrödinger Equations (Second Edition), IOP Publishing, ISBN 978-0-7503-5954-2. https://dx.doi.org/10.1088/978-0-7503-5954-2​
Amazon: https://a.co/d/2zWgL3v
Below is sample code for chapter 9 of the book.
CHAPTER 9: Nonlocal Nonlinear Schrödinger Equation

9.1 Introduction

Interest in the nonlocal NLSE has essentially started and then expanded significantly with the discovery of integrable class of nonlocal NLSE by Ablowitz and Musslimani [1]
i
∂
∂t
u(x,t)+
a
1
2
∂
∂
2
x
u(x,t)+
a
2
2
u
(x,t)
*
u
(-x,t)=0
(
1
)
Many other versions of the nonlocal NLSE have been considered [2] including discrete version of nonlocal NLSE [3]. It was also shown that the
nonlocal NLSE can be obtained by a gauge transformation of a coupled NLSE [4, 5], which may describe a magnetic system [4].
In a more general setting, nonlocal NLSE is obtained for optical materials with nonlocal nonlinear response,
i
∂
∂t
u(x,t)+
a
1
2
∂
∂
2
x
u(x,t)+
a
2
n(x,t)u(x,t)=0
(
2
)
where the response function,
G(x,t)
, defines the refractive index as
n(x,t)=
∞
∫
-∞
G(x-y)
2
u(y,t)
y
.In another setting, dipolar Bose-Einstein condensates are described by a nonlocal Gross-Pitaevskii equation [6]
iℏ
∂
∂t
ψ(r,t)=-
2
ℏ
2m
2
∇
ψ(r,t)+V(r)ψ(r,t)+g
2
ψ(r,t)
+
2
d

1-3
2
cos
θ
2
r-
′
r

2
ψ(
′
r
,t)

′
r
ψ(r,t)=0,
(
3
)
where
ψ(r,t)
is the wavefunction normalized to the number of atoms. The last term corresponds to the dipole-dipole interaction with strength,
d
, and angle between the dipole vectors,
θ
.
Since only the specific type of nonlocal NLSE given by Eq. (1) is integrable, while Eqs. (2) and (3) are not, only this version of the nonlocal NLSE including other variations such as coupled and discrete nonlocal NLSEs admit exact solutions.

How to use the code:

Each cell contains an exact solution and the equation it satisfies. Running the cell substitutes the solution into the equation and generates “0” as an output. Some cells contain functional transformations that transform one version of the nonlocal NLSE to another version, and hence the solutions of both equations will be linked through that transformation. These transformations are referred to as “Symmetry Reductions” in some cases.

9.2 Nonlocal NLSE

9.2.1 Scaling Transformation
From Local NLSE to Nonlocal NLSE

Finding solutions of the nonlocal NLSE:

Φ
t
+a1
Φ
xx
+a2
2
Φ
Φ
=0
from solutions of:

Ψ
t
+a1
Ψ
xx
+a2
2
Ψ
Ψ=0
where Ψ=Ψ[x,t;a1,a2],Φ=Φ[x,t;a1,a2],​
Φ
=
*
Φ
[-x,t;a1,a2],a1 and a2 are arbitrary real constants. ​​Transformation:
Φ[x,t;a1,a2]=
Ψ[x,t;a1,a2],
ifΨisanevenfunctioninx,
Ψ[x,t;a1,-a2],
ifΨisanoddfunctioninx.

Example 1. Even function: sech(x) bright soliton

The local solution:
ClearAll["Global`*"]​​ζ=x-x0;​​ψ[ζ_,t_]:=A0
2a1
a2
Sech[A0ζ]
(a1
2
A0
(t-t0)+ϕ0)

;​​ψs[ζ_,t_]:=A0
2a1
a2
Sech[A0ζ]
-(a1
2
A0
(t-t0)+ϕ0)

;​​​​eq=
(0,1)
ψ
[ζ,t]+a1
(2,0)
ψ
[ζ,t]+a2
2
ψ[ζ,t]
ψs[ζ,t];​​​​Simplify[eq]
0
A0, x0, t0, and ϕ0 are arbitrary real constants, a1 a2 > 0
The nonlocal local solution (same as above) :
ClearAll["Global`*"]​​ζ=x-x0;​​Φ[ζ_,t_]:=A0
2a1
a2
Sech[A0ζ]
(a1
2
A0
(t-t0)+ϕ0)

;​​Φs[ζ_,t_]:=A0
2a1
a2
Sech[A0ζ]
-(a1
2
A0
(t-t0)+ϕ0)

;​​​​eq=
(0,1)
Φ
[ζ,t]+a1
(2,0)
Φ
[ζ,t]+a2
2
Φ[ζ,t]
Φs[-ζ,t];​​​​Simplify[eq]
0

Example 2. Odd function: tanh(x) dark soliton


9.2 .2 Other Solutions


9.3 Yang’s Nonlocal NLSE

Equation:

Ψ
t
+a1
Ψ
xx
+a2
2
Ψ
+
2

Ψ

Ψ=0
,
where
​
Ψ
[x-
x
0
]=
*
Ψ
[-(x-
x
0
),t],
​
​
x
0
, a1 and a2 are arbitrary real constants.

Solution 1. Constant Amplitude Continuous wave (CW)

In[]:=
ClearAll["Global`*"]​​​​A1=2
2
A0
a2-a1
2
A2
;​​​​ψ[ζ_,t_]:=A0
(A1(t-t0)+A2(x-x0)+ϕ0)

​​ψs[ζ_,t_]:=A0
-(A1(t-t0)+A2(x-x0)+ϕ0)

​​ζ=x-x0;​​eq=D[ψ[ζ,t],t]+a1D[ψ[ζ,t],x,x]+a2(ψ[ζ,t]ψs[ζ,t]+ψ[-ζ,t]ψs[-ζ,t])ψ[ζ,t];​​(*A0,A2,x0,t0,andϕ0arearbitraryconstants*)​​​​x=3;t=2;​​Simplify[eq]
Out[]=
0

Solution 2.


Solution 3.


Solution 4.


9.4 Nonlocal Coupled NLSE

Finding solutions of the nonlocal coupled NLSE:

Φ
1
t
+
Φ
1
xx
+a1
Φ
1
Φ
1
+a2
Φ
2
Φ
2

Φ
1
=0,
​ ​

Φ
2
t
+
Φ
2
xx
+b1
Φ
1
Φ
1
+b2
Φ
2
Φ
2

Φ
2
=0
from solutions of:

Ψ
1
t
+
Ψ
1
xx
+(a1
Ψ
1
*
Ψ
1
+a2
Ψ
2
*
Ψ
2
)
Ψ
1
=0,
​ ​

Ψ
2
t
+
Ψ
2
xx
+(b1
Ψ
1
*
Ψ
1
+b2
Ψ
2
*
Ψ
2
)
Ψ
2
=0,
where ​
Ψ
1,2
=
Ψ
1,2
[x,t;a1,a2,b1,b2],​​
Φ
1,2
=
Φ
1,2
[x,t;a1,a2,b1,b2],​​
Φ
1,2
=
*
Φ
1,2
[-x,t;a1,a2,b1,b2],
​a1, a2, b1, and b2 are arbitrary real constants. ​​Transformation:
Φ
1
[x,t;a1,a2,b1,b2]=
Ψ
1
[x,t;a1,a2,b1,b2],
if
Ψ
1
isanevenfunctioninx,
Ψ
1
[x,t;-a1,a2,-b1,b2],
if
Ψ
1
isanoddfunctioninx,
​​​​
Φ
2
[x,t;a1,a2,b1,b2]=
Ψ
2
[x,t;a1,a2,b1,b2],
if
Ψ
2
isanevenfunctioninx,
Ψ
2
[x,t;a1,-a2,b1,-b2],
if
Ψ
2
isanoddfunctioninx.

Example 1. Even-Odd function: sech(x)-tanh(x) bright-dark soliton

The local solutions:
The nonlocal solutions:

9.5 Symmetry Reductions to Scalar Nonlocal NLSE

9.5.1 Symmetry Reduction I
From nonlocal Manakov system to scalar nonlocal NLSE

Finding solutions of:
from solutions of:

9.5.2 Symmetry Reduction II
From nonlocal Manakov system to scalar nonlocal NLSE

Finding solutions of:
from solutions of:

9.5.3 Symmetry Reduction III
From nonlocal vector NLSE to scalar nonlocal NLSE

Finding solutions of:
from solutions of:

9.6 Scaling Transformations

9.6.1 Linear and Nonlinear Coupling

9.6 .1 .1 General Case

Finding solutions of:
from solutions of:

9.6 .1 .2 Specific Case I: Nonlocal Manakov system to another nonlocal Manakov system

Finding solutions of:
from solutions of:

9.6 .1 .3 Specific Case II: Nonlocal Manakov system to the same nonlocal Manakov system
Superposition Principle for a Nonlocal Nonlinear System

Finding solutions of:

9.7 Nonlocal Discrete NLSE with Saturable Nonlinearity

Finding solutions of the nonlocal NLSE:
from solutions of:

9.7.1 Nonstaggered Solutions

Example 1. Even function: sech(n) discrete bright soliton

Solution of the local NLSE:
Solution of the nonlocal NLSE (same as above):

Bibliography

[1] Ablowitz M J and Musslimani Z H 2013. Integrable Nonlocal Nonlinear SchröNodinger Equation Phys. Rev. Lett. 110, 064105
[2] Ablowitz M J and Musslimani Z H 2016. Integrable Nonlocal Nonlinear Equations Studies in Applied Mathematics, Wiley Periodicals, Inc., A Wiley Company 139 7–59
[3] Ablowitz M J Xu-Dan Luo and Musslimani Z H 2020 Discrete nonlocal nonlinear SchröNodinger systems: Integrability, inverse scattering and solitons. Nonlinearity 33 3653
[4] Gadzhimuradov T A and Agalarov A M 2016. Towards a gauge-equivalent magnetic structure of the nonlocal nonlinear SchröNodinger equation Phys. Rev. A 93 062124
[5] Yang J 2018. Physically significant nonlocal nonlinear SchröNodinger equation and its soliton solutions. Phys. Rev. E 98 042202
[6] Santos L Shlyapnikov G V Zoller P and Lewenstein M 2000 Bose–Einstein condensation in trapped dipolar gases. Phys. Rev. Lett. 85 1791

CITE THIS NOTEBOOK

Handbook of exact solutions to the nonlinear Schrödinger equations​
by Usama Al Khawaja & Laila Al Sakkaf​
Wolfram Community, STAFF PICKS, November 19, 2024
​https://community.wolfram.com/groups/-/m/t/3323786