Handbook of exact solutions to the nonlinear Schrodinger equations / Usama Al Khawaja and Laila Al Sakkaf.

Format:

Book

Author/Creator:

Al Khawaja, Usama, author.

Al Sakkaf, Laila, author.

Series:

IOP Ebooks Series

Language:

English

Subjects (All):

Nonlinear integral equations.

Physical Description:

1 online resource (451 pages)

Edition:

Second edition.

Place of Publication:

Bristol, England : IOP Publishing, [2024]

Summary:

This book aims to organise and centralise existing solutions of the nonlinear Schrödinger equation (NLSE). 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.

Contents:

Intro

Preface to First Edition

Preface to Second Edition

Notation

Acknowledgements

Author Biographies

Usama Al Khawaja

Laila Al Sakkaf

Chapter Introduction

References

Chapter Fundamental Nonlinear Schrödinger Equation

A Glance at Chapter 2

A Statistical View of Chapter 2

2.1 Introduction

2.1.1 Main Features of NLSE

2.1.2 Generic Derivation

2.1.3 Bose-Einstein Condensates

2.1.4 Light Pulses in Optical Fibers

2.1.5 Surface Water Waves

2.2 NLSE with Cubic Nonlinearity

2.2.1 Real Dispersion and Nonlinearity Coefficients

2.2.2 Summary of Section 2.2.1

2.2.3 Complex Dispersion and Nonlinearity Coefficients

2.2.4 Summary of Section 2.2.3

Chapter Nonlinear Schrödinger Equation with Power Law and Dual Power law Nonlinearities

A Glance at Chapter 3

A Statistical View of Chapter 3

3.1 Introduction

3.2 NLSE with Power Law Nonlinearity

3.2.1 Reduction to the Fundamental NLSE

3.3 Summary of Section 3.2

3.4 NLSE with Dual Power Law Nonlinearity

3.5 Summary of Section 3.4

Chapter Nonlinear Schrödinger Equation with Higher Order Terms

A Glance at Chapter 4

A Statistical View of Chapter 4

4.1 Introduction

4.1.1 Optical Pulses

4.1.2 Bose-Einstein Condensates

4.1.3 Water Surface Waves

4.2 NLSE with Third Order Dispersion, Self-Steepening, and Self-Frequency Shift

4.3 Summary of Section 4.2

4.4 Special Cases of Equation (4.17)

4.4.1 Case I: Hirota Equation (HE)

4.4.2 Case II: Sasa-Satsuma Equation (SSE)

4.5 NLSE with First and Third Order Dispersions, Self-Steepening, Self-Frequency Shift, and Potential

4.6 Summary of Section 4.5

4.7 NLSE with t-dependent Coefficients and First Order Dispersion

4.8 Summary of Section 4.7

4.9 NLSE with Fourth Order Dispersion

4.10 Summary of Section 4.9.

4.11 NLSE with Fourth Order Dispersion and Power Law Nonlinearity

4.12 Summary of Section 4.11

4.13 NLSE with Third and Fourth Order Dispersions and Cubic and Quintic Nonlinearities

4.14 Summary of Section 4.13

4.15 NLSE with Third and Fourth Order Dispersions, Self-Steepening, Self-Frequency Shift, and Cubic and Quintic Nonlinearities

4.16 Summary of Section 4.15

4.17 NLSE with ∣ψ∣2-Dependent Dispersion

4.18 Infinite Hierarchy of Integrable NLSEs with Higher Order Terms

4.18.1 Constant Coefficients

4.18.2 Function Coefficients

4.19 Summary of Section 4.18

Chapter Scaling Transformations

A Glance at Chapter 5

A Statistical View of Chapter 5

5.1 Introduction

5.2 Fundamental NLSE to Fundamental NLSE with Different Constant Coefficients

5.3 Defocusing (Focusing) NLSE to Focusing (Defocusing) NLSE

5.4 Galilean Transformation (Moving Solutions)

5.5 Function Coefficients

5.5.1 Constant Dispersion and Complex Potential

5.5.2 Constant Dispersion and Real Quadratic Potential

5.5.3 Constant Dispersion and Real Linear Potential

5.5.4 Constant Nonlinearity and Complex Potential

5.5.5 Constant Nonlinearity and Real Quadratic Potential

5.5.6 Constant Nonlinearity and Real Linear Potential

5.6 Solution-Dependent Transformation

5.6.1 Special Case 1: Stationary Solution, Constant Dispersion and Nonlinearity Coefficients

5.6.2 Special Case 2: PT-Symmetric Potential

5.6.3 Special Case 3: Stationary Solution, Constant Dispersion and Nonlinearity Coefficients, and Real Potential

5.7 Summary of Sections 5.2-5.6

5.8 Other Equations

5.8.1 NLSE with Periodic Potentials

5.8.2 NLSE with Pöschl-Teller Potential

5.9 Summary of Section 5.8

Chapter Nonlinear Schrödinger Equation in (N + 1)-Dimensions

A Glance at Chapter 6.

A Statistical View of Chapter 6

6.1 Introduction

6.2 (N + 1)-Dimensional NLSE with Cubic Nonlinearity

6.3 (N + 1)-Dimensional NLSE with Power Law Nonlinearity

6.4 (N + 1)-Dimensional NLSE with Dual Power Law Nonlinearity

6.5 Galilean Transformation in (N + 1)-Dimensions (Moving Solutions)

6.6 NLSE in (2 + 1)-Dimensions with Φx1x2 Term

6.7 Summary of Sections 6.2-6.6

6.8 (N + 1)-Dimensional Isotropic NLSE with Cubic Nonlinearity in Polar Coordinate System

6.8.1 Angular Dependence

6.8.2 Constant Dispersion and Real Potential

6.9 Summary of Section 6.8

6.10 Power Series Solutions to (2 + 1)-Dimensional NLSE with Cubic Nonlinearity in Polar Coordinate System

6.10.1 Family of Infinite Number of Localized Solutions

Chapter Coupled Nonlinear Schrödinger Equations

A Glance at Chapter 7

A Statistical View of Chapter 7

7.1 Introduction

7.1.1 Single-Mode Optical Fibers

7.1.2 Coupled Modes Optical Fibers

7.1.3 Mixtures and Spinor Bose-Einstein Condensates

7.2 Fundamental Coupled NLSE Manakov System

7.3 Summary of Section 7.2

7.4 Symmetry Reductions

7.4.1 Symmetry Reduction I

7.4.2 Symmetry Reduction II

7.4.3 Symmetry Reduction III

7.4.4 Symmetry Reduction IV

7.4.5 Symmetry Reduction V

7.5 Scaling Transformations

7.5.1 Linear and Nonlinear Coupling

7.5.2 Complex Coupling

7.5.3 Function Coefficients

7.6 Summary of Sections 7.4-7.5

7.7 (N + 1)-Dimensional Coupled NLSE (N + 1)-Dimensional Manakov System

7.7.1 Reduction to 1D Manakov System

7.8 Symmetry Reductions of (N + 1)-Dimensional CNLSE to Scalar NLSE

7.8.1 Symmetry Reduction I From (N + 1)-Dimensional Manakov System to (N + 1)-Dimensional Fundamental NLSE

7.8.2 Symmetry Reduction II From (N + 1)-Dimensional Manakov System to (N + 1)-Dimensional Fundamental NLSE.

7.8.3 Symmetry Reduction III From (N+1)-Dimensional Vector NLSE to (N + 1)-Dimensional Fundamental NLSE

7.9 (N + 1)-Dimensional Scaling Transformations

7.9.1 Linear and Nonlinear Coupling

7.9.2 Complex Coupling

7.10 Composite Solutions: Nonlinear Superposition

7.11 Summary of Sections 7.8-7.10

Chapter Discrete Nonlinear Schrödinger Equation

A Glance at Chapter 8

A Statistical View of Chapter 8

8.1 Introduction

8.2 Discrete NLSE with Saturable Nonlinearity

8.2.1 Nonstaggered Solutions

8.2.2 Staggered Solutions

8.3 Summary of Section 8.2

8.4 Short-period Solutions with General, Kerr, and Saturable Nonlinearities

8.5 Ablowitz-Ladik Equation

8.6 Summary of Section 8.5

8.7 Cubic-Quintic Discrete NLSE

8.8 Summary of Section 8.7

8.9 Generalized Discrete NLSE

8.10 Summary of Section 8.9

8.11 Coupled Salerno Equations

8.12 Summary of Section 8.11

8.13 Coupled Ablowitz-Ladik Equation

8.14 Summary of Section 8.13

8.15 Coupled Saturable Discrete NLSE

8.16 Summary of Section 8.15

Chapter Nonlocal Nonlinear Schrödinger Equation

A Glance at Chapter 9

A Statistical View of Chapter 9

9.1 Introduction

9.2 Nonlocal NLSE

9.2.1 Scaling Transformation From Local NLSE to Nonlocal NLSE

9.2.2 Other Solutions

9.3 Yang's Nonlocal NLSE

9.4 Nonlocal Coupled NLSE

9.5 Symmetry Reductions to Scalar Nonlocal NLSE

9.5.1 Symmetry Reduction I From Nonlocal Manakov System to Scalar Nonlocal NLSE

9.5.2 Symmetry Reduction II From Nonlocal Manakov System to Scalar Nonlocal NLSE

9.5.3 Symmetry Reduction III From Nonlocal Vector NLSE to Scalar Nonlocal NLSE

9.6 Scaling Transformations

9.6.1 Linear and Nonlinear Coupling

9.6.2 Complex Coupling

9.7 Nonlocal Discrete NLSE with Saturable Nonlinearity

9.7.1 Nonstaggered Solutions.

9.7.2 Staggered Solutions

9.8 Nonlocal Ablowitz-Ladik Equation

9.9 Nonlocal Cubic-Quintic Discrete NLSE

9.10 Summary of Chapter 9

Chapter Fractional Nonlinear Schrödinger Equation

A Glance at Chapter 10

A Statistical View of Chapter 10

10.1 Introduction

10.2 Conformable Fractional Derivative

10.2.1 NLSE with Conformable Space-Time Fractional Derivative

10.2.2 Resonant NLSE with Conformable Time Fractional Derivative

10.2.3 Resonant Cubic-Quintic NLSE with Conformable Time Fractional Derivative

10.2.4 NLSE with Conformable Space-Time Fractional Derivative, First Order Spatial Dispersion, and Second Order Temporal Dispersion

10.3 Summary of Section 10.2

10.4 Fractal Fractional Derivative

10.4.1 NLSE with Fractal Space-Time Fractional Derivative

10.5 Summary of Section 10.4

10.6 Atangana (β) Fractional Derivative

10.6.1 NLSE with Atangana Space-Time Fractional Derivative

10.7 Summary of Section 10.6

Chapter

A.1 Derivation of Some Solutions of Section 2.2

A.1.1 Schematic Representation

A.1.2 Detailed Derivations

A.2 Derivation of Some Solutions of Section 3.2

A.2.1 Schematic Representation

A.2.2 Detailed Derivations

A.3 Derivation of Some Solutions of Section 3.4

A.3.1 Schematic Representation

A.3.2 Detailed Derivations

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

C.1 Function Coefficients

C.2 Solution-Dependent Transformation

C.3 Similarity Transformation for the NLSE in (N + 1)-Dimensions.

Notes:

Description based on publisher supplied metadata and other sources.

Description based on print version record.

Includes bibliographical references.

ISBN:

9780750359535

0750359536

9780750359542

0750359544

OCLC:

1460467125