Nonlinear waves & Hamiltonian systems : from one to many degrees of freedom, from discrete to continuum / R. Carretero-González, D.J. Frantzeskakis, P.G. Kevrekidis.

Format:

Book

Author/Creator:

Carretero-González, Ricardo, author.

Frantzeskakis, Dimitri J., author.

Kevrekidis, Panayotis G., author.

Series:

Oxford scholarship online.

Oxford scholarship online

Language:

English

Subjects (All):

Nonlinear wave equations.

Hamiltonian systems.

Physical Description:

1 online resource (561 pages)

Edition:

1st ed.

Place of Publication:

Oxford : Oxford University Press, [2024]

Summary:

The aim of this book is to provide a self-contained introduction to the continuously developing field of nonlinear waves, that offers the background, the basic ideas and mathematical, as well as computational methods, while also presenting an overview of associated physical applications.

Contents:

Cover

Title page

Copyright page

Preface

Acknowledgments

Contents

PART I INTRODUCTION AND MOTIVATION OF MODELS

1 Introduction and Motivation

1.1 Few degrees of freedom

A linear example: Hooke's law

A nonlinear example: the pendulum

1.2 Many degrees of freedom

1.3 A nonlinear variant: the FPUT lattice

Exercises

2 Linear Dispersive Wave Equations

2.1 Dispersion relations and relevant notions

2.2 Examples of linear dispersive wave equations

The transport (unidirectional wave) equation

The (bidirectional) wave equation

The linear Schrödinger equation

2.3 Wavepackets and group velocity

2.4 Dissipation, instability, and diffusion

3 Nonlinear Dispersive Wave Equations

3.1 Dispersion relations, linear and nonlinear equations

3.2 Unidirectional propagation: KdV, KP, and NLS

The Korteweg-de Vries (KdV) equation

Other versions of the KdV model

The Kadomtsev-Petviashvili (KP) equation

The nonlinear Schrödinger equation

3.3 Bidirectional propagation: KG and Boussinesq

The Klein-Gordon equation

The Boussinesq equation

PART II KORTEWEG-DE VRIES (KDV) EQUATION

4 The Korteweg-de Vries (KdV) Equation

4.1 Obtaining KdV as a limit of FPUT

4.2 Obtaining KdV for shallow water waves

4.3 The effects of dispersion and nonlinearity

The effect of dispersion - linearized KdV equation

The effect of nonlinearity - Hopf equation, method of characteristics and shock waves

4.4 Putting it all together: the Zabusky-Kruskal numerical experiments

4.5 A cute twist: conservation laws

5 From Boussinesq to KdV - Boussinesq Solitons as KdV Solitons

5.1 Boussinesq to a single KdV (right-going waves)

5.2 Boussinesq to a KdV pair (right- and left-going waves)

5.3 Connecting the Boussinesq soliton with the KdV soliton.

5.4 A higher dimensional generalization: Boussinesq to KP

6 Traveling Wave Reduction, Elliptic Functions, and Connections to KdV

6.1 Traveling wave reduction: the quadrature problem

6.2 Elliptic function solutions

6.3 Another traveling wave reduction example: the Boussinesq equation

7 Burgers and KdV-Burgers Equations - Regularized Shock Waves

7.1 Shock waves of the Burgers equation

The traveling shock wave

Structure of the traveling shock

7.2 The Cole-Hopf transformation

Introducing the transformation

Solution of the viscous Burgers equation

Comments on the Cole-Hopf transformation

7.3 The KdVB equation - a model for dispersive shocks

Traveling waves and fixed points in the associated phase plane

Dissipation vs. dispersion - monotonic vs. oscillating shocks

8 A Final Touch From KdV: Invariances and Self-Similar Solutions

8.1 Scale and Galilean invariances

Scale invariance

Galilean invariance

8.2 The KdV self-similar waveforms

8.3 Going beyond the KdV: seeking self-similarity

8.4 A familiar example with a little-known twist: the diffusion equation

9 Spectral Methods

9.1 Revisiting the Fourier transform and the golden property

9.2 Spectral methods for PDEs

Preliminaries: the discrete Fourier transform

Example 1: the variable speed, linear, transport equation

Example 2: split-step spectral integration: the KdV equation

10 Bäcklund Transformation for the KdV

10.1 The general idea and a simple introductory example: Laplace equation

10.2 Bäcklund transformation for the KdV

11 Inverse Scattering Transform I - the KdV Equation*

11.1 Introduction and motivation

11.2 Description of the Inverse Scattering Transform method

11.3 The linear problem associated with the KdV equation.

Connection of the KdV with the Schrödinger equation

The scattering problem for the Schrödinger equation

11.4 The problem of the evolution of the scattering data

A closer look at the Lax operators

The operator M for the KdV equation

Time evolution of the scattering data

11.5 The inverse problem - Gel'fand-Levitan-Marchenko (GLM) equation

11.6 Soliton solutions of the KdV equation

The one-soliton solution

The two-soliton solution

The collision between two solitons

Generalization: the N-soliton solution

12 Direct Perturbation Theory for Solitons*

12.1 General aspects of the perturbative approach

12.2 KdV with a dissipative perturbation

12.3 Transverse instability of line solitons of the KP equation

13 The Kadomtsev-Petviashvili Equation*

13.1 The KP equation and its variants

13.2 Basic properties of the KP equation

13.3 Line solitons and lumps of the KP equation

13.4 Interactions of line solitons of the KP-II equation

13.5 Finite-genus and quasi-periodic solutions

PART III KLEIN-GORDON, SINE-GORDON, AND PHI-4 MODELS

14 Another Class of Models: Nonlinear Klein-Gordon Equations

14.1 Models, physical motivation, and principal kink waveforms

14.2 Linear and nonlinear transformations

Linear transformations: traveling kinks

Nonlinear transformations: breathers

14.3 Linear stability, homogeneous steady states, linear dispersion relations

14.4 Linearization around non-homogeneous states: kinks

14.5 A stability tool: the Evans function

The Evans function: introduction

The Evans function: an example

The Evans function: final remarks

15 Additional Tools/Results for Klein-Gordon Equations

15.1 Finding solitons using the method of B¨acklund transforms

15.2 Examining soliton interactions: the Manton method.

15.3 Kink-kink and kink-antikink collisions

15.4 Higher dimensional sine-Gordon equation: a teaser and an invitation

15.5 Sine-Gordon with gain and loss: a cute application

16 Klein-Gordon to NLS Connection - Breathers as NLS Solitons

16.1 Preliminaries: model and the method of multiple scales

16.2 NLS from Klein-Gordon

16.3 Sine-Gordon breathers as NLS bright solitons

16.4 On the formal derivation and universality of NLS

17 Interlude: Numerical Considerations for Nonlinear Wave Equations

17.1 Existence and stability

17.2 Nonlinear dynamics

PART IV THE NONLINEAR SCHR¨ODINGER EQUATIONS

18 The Nonlinear Schr¨odinger (NLS) Equation

18.1 Obtaining linear and nonlinear Schr¨odinger from dispersive wavepackets

18.2 Plane wave solutions and modulational instability

18.3 A more general analysis: solitons and periodic solutions

19 NLS to KdV Connection - Dark Solitons as KdV Solitons

19.1 Preliminaries and motivation: structure of the dark soliton

19.2 Defocusing NLS to Boussinesq and to KdV

Model and linear theory

NLS to Boussinesq and to two KdVs

19.3 Direct derivation of the KdV

19.4 Shallow dark solitons vs. KdV solitons

20 Actions, Symmetries, Conservation Laws, Noether's Theorem, and All That

20.1 Lagrangian and Hamiltonian formalisms

20.2 Poisson brackets

20.3 Noether's theorem

20.4 Field-theoretic variants

20.5 The nonlinear Schrödinger case example

20.6 Connections with the cubic NLS and its soliton linearization problem

20.7 Connections with wave collapse: NLS with generalized power law nonlinearity

20.8 Final twist: antisymmetric operators and generalized Poisson brackets

21 Applications of Conservation Laws - Adiabatic Perturbation Method.

21.1 Scaling and phase invariances and Galilean boost for NLS solutions

21.2 Conservation laws and conserved quantities for NLS: redux

21.3 Adiabatic perturbation theory for bright solitons

Perturbation-induced evolution of the soliton parameters

Application I: bright solitons under linear loss

Application II: bright solitons in external potentials

21.4 Adiabatic perturbation theory for dark solitons

Renormalized conserved quantities

Evolution of the soliton parameters

Application: gain with saturation

22 Numerical Techniques for NLS

22.1 Finite differences

22.2 Steady states: Newton's method

22.3 Stability

22.4 Dynamics: finite differences and RK4

22.5 Example: soliton on top of a hill

22.6 Validating your codes and results

23 Inverse Scattering Transform II - the NLS Equation*

23.1 The Ablowitz-Kaup-Newell-Segur (AKNS) approach

Introducing the matrix formulation

The AKNS pair for the NLS equation

The AKNS family: a wealth of integrable equations

23.2 IST for the focusing NLS equation

The scattering problem - direct transform

The inverse transform

23.3 Soliton solutions

23.4 A note on the defocusing NLS and the role of boundary conditions

23.5 The inverse problem as a Riemann-Hilbert problem.

Basics on Riemann-Hilbert problems

Application to the focusing NLS equation

24 The Gross-Pitaevskii (GP) Equation

24.1 Physical motivation: Bose-Einstein condensates

24.2 The Gross-Pitaevskii (GP) equation

24.3 Dimensional reductions and adimensionalization

3D to 2D reduction

3D to 1D reduction

Adimensionalization

24.4 Small and large mass limits

Small mass limit: the linear limit.

Large mass limit: the Thomas-Fermi approximation.

Notes:

Includes bibliographical references and index.

Description based on online resource and publisher information; title from PDF title page (viewed on October 9, 2024).

ISBN:

9780198903109

0198903103

9780192654946

0192654942

OCLC:

1460295776