This book collects all known solutions to the nonlinear Schrödinger equation in one resource. In addition, the book organizes the solutions by classifying and grouping them based on aspects and symmetries they possess. Accompaniedby Mathematica Notebooks containing all solutions and features a large number of figures and animations.
Author(s): Usama Al Khawaja, Laila Al Sakkaf
Publisher: IOP Publishing
Year: 2020
Language: English
Pages: 375
PRELIMS.pdf
Preface
Acknowledgments
Author Biographies
Usama Al Khawaja
Laila Al Sakkaf
Notation
CH001.pdf
Chapter 1 Introduction
References
CH002.pdf
Chapter 2 Fundamental Nonlinear Schrödinger Equation
A Glance at Chapter 2
A Statistical View of Chapter 2
2.1 NLSE with Cubic Nonlinearity
2.1.1 Real Dispersion and Nonlinearity Coefficients
2.2 Summary of Subsection 2.1.1
2.2.1 Complex Dispersion and Nonlinearity Coefficients
2.3 Summary of Subsection 2.2.1
References
CH003.pdf
Chapter 3 Nonlinear Schrödinger Equation with Power Law and Dual Power Law Nonlinearities
A Glance at Chapter 3
A Statistical View of Chapter 3
3.1 NLSE with Power Law Nonlinearity
3.1.1 Reduction to the Fundamental NLSE
3.2 Summary of Section 3.1
3.3 NLSE with Dual Power Law Nonlinearity
3.4 Summary of Section 3.3
References
CH004.pdf
Chapter 4 Nonlinear Schrödinger Equation with Higher Order Terms
A Glance at Chapter 4
A Statistical View of Chapter 4
4.1 NLSE with Third Order Dispersion, Self-Steepening, and Self-Frequency Shift
4.2 Summary of Section 4.1
4.3 Special Cases of Equation (4.1)
4.3.1 Case I: Hirota Equation (HE)
4.3.2 Case II: Sasa–Satsuma Equation (SSE)
4.4 NLSE with First and Third Order Dispersions, Self-Steepening, Self-Frequency Shift, and Potential
4.5 Summary of Section 4.4
4.6 NLSE with Fourth Order Dispersion
4.7 Summary of Section 4.6
4.8 NLSE with Fourth Order Dispersion and Power Law Nonlinearity
4.9 Summary of Section 4.8
4.10 NLSE with Third and Fourth Order Dispersions and Cubic and Quintic Nonlinearities
4.11 Summary of Section 4.10
4.12 NLSE with Third and Fourth Order Dispersions, Self-Steepening, Self-Frequency Shift, and Cubic and Quintic Nonlinearities
4.13 Summary of Section 4.12
4.14 NLSE with ∣ψ∣2-Dependent Dispersion
4.15 Infinite Hierarchy of Integrable NLSEs with Higher Order Terms
4.15.1 Constant Coefficients
4.15.2 Function Coefficients
4.16 Summary of Section 4.15
References
CH005.pdf
Chapter 5 Scaling Transformations
A Glance at Chapter 5
A Statistical View of Chapter 5
5.1 Fundamental NLSE to Fundamental NLSE with Different Constant Coefficients
5.2 Defocusing (Focusing) NLSE to Focusing (Defocusing) NLSE
5.3 Galilean Transformation (Movable Solutions)
5.4 Function Coefficients
5.4.1 Constant Dispersion and Complex Potential
5.4.2 Constant Dispersion and Real Quadratic Potential
5.4.3 Constant Dispersion and Real Linear Potential
5.4.4 Constant Nonlinearity and Complex Potential
5.4.5 Constant Nonlinearity and Real Quadratic Potential
5.4.6 Constant Nonlinearity and Real Linear Potential
5.5 Solution-Dependent Transformation
5.5.1 Special Case I: Stationary Solution, Constant Dispersion and Nonlinearity Coefficients
5.5.2 Special Case II: PT-Symmetric Potential
5.5.3 Special Case III: Stationary Solution, Constant Dispersion and Nonlinearity Coefficients, and Real Potential
5.6 Summary of Sections 5.1–5.5
5.7 Other Equations: NLSE with Periodic Potentials
5.7.1 General Case: sn2(x,m) Potential
5.7.2 Specific Case: sin2(x) Potential
5.8 Summary of Section 5.7
Reference
CH006.pdf
Chapter 6 Nonlinear Schrödinger Equation in (N + 1)-Dimensions
A Glance at Chapter 6
A Statistical View of Chapter 6
6.1 (N + 1)-Dimensional NLSE with Cubic Nonlinearity
6.2 (N + 1)-Dimensional NLSE with Power Law Nonlinearity
6.3 (N + 1)-Dimensional NLSE with Dual Power Law Nonlinearity
6.4 Galilean Transformation in (N + 1)-Dimensions (Movable Solutions)
6.5 NLSE in (2 + 1)-Dimensions with Φx1x2 Term
6.6 Summary of Sections 6.1–6.5
6.7 (N + 1)-Dimensional Isotropic NLSE with Cubic Nonlinearity in Polar Coordinate System
6.7.1 Angular Dependence
6.7.2 Constant Dispersion and Real Potential
6.8 Summary of Section 6.7
6.9 Power Series Solutions to (2 + 1)-Dimensional NLSE with Cubic Nonlinearity in a Polar Coordinate System
6.9.1 Family of Infinite Number of Localized Solutions
References
CH007.pdf
Chapter 7 Coupled Nonlinear Schrödinger Equations
A Glance at Chapter 7
A Statistical View of Chapter 7
7.1 Fundamental Coupled NLSE Manakov System
7.2 Summary of Section 7.1
7.3 Symmetry Reductions
7.3.1 Symmetry Reduction I
7.3.2 Symmetry Reduction II
7.3.3 Symmetry Reduction III
7.3.4 Symmetry Reduction IV
7.3.5 Symmetry Reduction V
7.4 Scaling Transformations
7.4.1 Linear and Nonlinear Coupling
7.4.2 Complex Coupling
7.4.3 Function Coefficients
7.5 Summary of Sections 7.3–7.4
7.6 (N + 1)-Dimensional Coupled NLSE
7.6.1 Reduction to 1D Manakov System
7.7 Symmetry Reductions of (N + 1)-Dimensional CNLSE to Scalar NLSE
7.7.1 Symmetry Reduction I
7.7.2 Symmetry Reduction II
7.7.3 Symmetry Reduction III
7.8 (N + 1)-Dimensional Scaling Transformations
7.8.1 Linear and Nonlinear Coupling
7.8.2 Complex Coupling
7.9 Summary of Sections 7.7–7.8
References
CH008.pdf
Chapter 8 Discrete Nonlinear Schrödinger Equation
A Glance at Chapter 8
A Statistical View of Chapter 8
8.1 Discrete NLSE with Saturable Nonlinearity
8.1.1 Nonstaggered Solutions
8.1.2 Staggered Solutions
8.2 Summary of Section 8.1
8.3 Short-period Solutions with General, Kerr, and Saturable Nonlinearities
8.4 Ablowitz–Ladik Equation
8.5 Summary of Section 8.4
8.6 Cubic-quintic Discrete NLSE
8.7 Summary of Section 8.6
8.8 Generalized Discrete NLSE
8.9 Summary of Section 8.8
8.10 Coupled Salerno Equations
8.11 Summary of Section 8.10
8.12 Coupled Ablowitz–Ladik Equation
8.13 Summary of Section 8.12
8.14 Coupled Saturable Discrete NLSE
8.15 Summary of Section 8.14
References
CH009.pdf
Chapter 9 Nonlocal Nonlinear Schrödinger Equation
A Glance at Chapter 9
A Statistical View of Chapter 9
9.1 Nonlocal NLSE
9.2 Nonlocal Coupled NLSE
9.3 Symmetry Reductions to Scalar Nonlocal NLSE
9.3.1 Symmetry Reduction I
9.3.2 Symmetry Reduction II
9.3.3 Symmetry Reduction III
9.4 Scaling Transformations
9.4.1 Linear and Nonlinear Coupling
9.4.2 Complex Coupling
9.5 Nonlocal Discrete NLSE with Saturable Nonlinearity
9.5.1 Nonstaggered Solutions
9.5.2 Staggered Solutions
9.6 Nonlocal Ablowitz–Ladik Equation
9.7 Nonlocal Cubic-Quintic Discrete NLSE
9.8 Summary of Chapter 9
APP1.pdf
Chapter
A.1 Derivation of Some Solutions of Section 2.1
A.1.1 Schematic Representation
A.1.2 Detailed Derivations
A.2 Derivation of Some Solutions of Section 3.1
A.2.1 Schematic Representation
A.2.2 Detailed Derivations
A.3 Derivation of Some Solutions of Section 3.3
A.3.1 Schematic Representation
A.3.2 Detailed Derivations
APP2.pdf
Chapter
B.1 Darboux Transformation
Link between NLSE and LP
Seed Solution
Darboux Transformation
Symmetry Reduction
B.1.1 Bright Soliton Solution: Zero Seed
B.1.2 Generalized Breather Solution for Focusing and Defocusing Nonlinearity: CW Seed
APP3.pdf
Chapter
C.1 Function Coefficients
C.2 Solution-Dependent Transformation
C.3 Similarity Transformation for the NLSE in (N + 1)-Dimensions
