(2025/2026): Exact-Solution Methods for the Nonlinear Schrödinger Equation

, Contents

Preface x
Acknowledgments xii
Author Biographies xiii
Notation xiv

1 Introduction 1-1
References 1-6

2 Fundamental Nonlinear Schrödinger Equation 2-1
2.1 NLSE with Cubic Nonlinearity 2-1
2.1.1 Real Dispersion and Nonlinearity Coefficients 2-2
2.2 Summary of Subsection 2.1.1 2-33
2.2.1 Complex Dispersion and Nonlinearity Coefficients 2-40
2.3 Summary of Subsection 2.2.1 2-43
References 2-45

3 Nonlinear Schrödinger Equation with Power 3-1
Law and Dual Power Law Nonlinearities
3.1 NLSE with Power Law Nonlinearity 3-1
3.1.1 Reduction to the Fundamental NLSE 3-2
3.2 Summary of Section 3.1 3-6
3.3 NLSE with Dual Power Law Nonlinearity 3-8
3.4 Summary of Section 3.3 3-14
References 3-17

4 Nonlinear Schrödinger Equation with Higher Order Terms 4-1
4.1 NLSE with Third Order Dispersion, Self- 4-3
Steepening, and Self-Frequency Shift
4.2 Summary of Section 4.1 4-9
4.3 Special Cases of Equation (4.1) 4-13
4.3.1 Case I: Hirota Equation (HE) 4-13
4.3.2 Case II: Sasa–Satsuma Equation (SSE) 4-13
4.4 NLSE with First and Third Order Dispersions, Self- 4-13
Steepening, Self-Frequency Shift, and Potential



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4.5 Summary of Section 4.4 4-16
4.6 NLSE with Fourth Order Dispersion 4-17
4.7 Summary of Section 4.6 4-19
4.8 NLSE with Fourth Order Dispersion and Power Law Nonlinearity4-20
4.9 Summary of Section 4.8 4-22
4.10 NLSE with Third and Fourth Order Dispersions and 4-24
Cubic and Quintic Nonlinearities
4.11 Summary of Section 4.10 4-29
4.12 NLSE with Third and Fourth Order Dispersions, Self- 4-32
Steepening, Self-Frequency Shift, and Cubic and Quintic
Nonlinearities
4.13 Summary of Section 4.12 4-36
4.14 NLSE with ∣ψ∣2-Dependent Dispersion 4-39
4.15 Infinite Hierarchy of Integrable NLSEs with Higher Order Terms4-40
4.15.1 Constant Coefficients 4-40
4.15.2 Function Coefficients 4-43
4.16 Summary of Section 4.15 4-46
References 4-49

5 Scaling Transformations 5-1
5.1 Fundamental NLSE to Fundamental NLSE 5-4
with Different Constant Coefficients
5.2 Defocusing (Focusing) NLSE to Focusing (Defocusing) NLSE 5-5
5.3 Galilean Transformation (Movable Solutions) 5-6
5.4 Function Coefficients 5-10
5.4.1 Constant Dispersion and Complex Potential 5-10
5.4.2 Constant Dispersion and Real Quadratic Potential 5-11
5.4.3 Constant Dispersion and Real Linear Potential 5-18
5.4.4 Constant Nonlinearity and Complex Potential 5-24
5.4.5 Constant Nonlinearity and Real Quadratic Potential 5-25
5.4.6 Constant Nonlinearity and Real Linear Potential 5-25
5.5 Solution-Dependent Transformation 5-26
5.5.1 Special Case I: Stationary Solution, Constant 5-27
Dispersion and Nonlinearity Coefficients
5.5.2 Special Case II: PT-Symmetric Potential 5-28
5.5.3 Special Case III: Stationary Solution, Constant 5-29
Dispersion and Nonlinearity Coefficients, and Real
Potential
5.6 Summary of Sections 5.1–5.5 5-30
5.7 Other Equations: NLSE with Periodic Potentials 5-38

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5.7.1 General Case: sn2(x, m) Potential 5-38
5.7.2 Specific Case: sin2(x) Potential 5-39
5.8 Summary of Section 5.7 5-40
Reference 5-40

6 Nonlinear Schrödinger Equation in (N + 1)-Dimensions 6-1
6.1 (N + 1)-Dimensional NLSE with Cubic Nonlinearity 6-4
6.2 (N + 1)-Dimensional NLSE with Power Law Nonlinearity 6-11
6.3 (N + 1)-Dimensional NLSE with Dual Power Law Nonlinearity 6-12
6.4 Galilean Transformation in (N + 1)-Dimensions (Movable Solutions)6-16
6.5 NLSE in (2 + 1)-Dimensions with Φx1x2 Term 6-22
6.6 Summary of Sections 6.1–6.5 6-24
6.7 (N + 1)-Dimensional Isotropic NLSE with Cubic 6-33
Nonlinearity in Polar Coordinate System
6.7.1 Angular Dependence 6-34
6.7.2 Constant Dispersion and Real Potential 6-35
6.8 Summary of Section 6.7 6-38
6.9 Power Series Solutions to (2 + 1)-Dimensional NLSE with 6-41
Cubic Nonlinearity in a Polar Coordinate System
6.9.1 Family of Infinite Number of Localized Solutions 6-42
References 6-42

7 Coupled Nonlinear Schrödinger Equations 7-1
7.1 Fundamental Coupled NLSE Manakov System 7-4
7.2 Summary of Section 7.1 7-13
7.3 Symmetry Reductions 7-17
7.3.1 Symmetry Reduction I From Manakov 7-17
System to Fundamental NLSE
7.3.2 Symmetry Reduction II From Manakov 7-17
System to Fundamental NLSE
7.3.3 Symmetry Reduction III From Vector 7-18
NLSE to Fundamental NLSE
7.3.4 Symmetry Reduction IV From Three Coupled 7-19
NLSEs to Manakov System
7.3.5 Symmetry Reduction V From Vector
7-22
NLSE to Manakov System
7.4 Scaling Transformations 7-22
7.4.1 Linear and Nonlinear Coupling 7-22
7.4.2 Complex Coupling 7-25



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