Nonlinear periodic waves and their modulations : an introductory course / A.M. Kamchatnov.

Format:

Book

Author/Creator:

Kamchatnov, A. M. (Anatoliĭ Mikhaĭlovich)

Language:

English

Subjects (All):

Nonlinear waves.

Wave-motion, Theory of.

Solitons.

Physical Description:

1 online resource (399 p.)

Place of Publication:

Singapore ; River Edge, NJ : World Scientific, c2000.

Language Note:

English

Summary:

Although the mathematical theory of nonlinear waves and solitons has made great progress, its applications to concrete physical problems are rather poor, especially when compared with the classical theory of linear dispersive waves and nonlinear fluid motion. The Whitham method, which describes the combining action of the dispersive and nonlinear effects as modulations of periodic waves, is not widely used by applied mathematicians and physicists, though it provides a direct and natural way to treat various problems in nonlinear wave theory. Therefore it is topical to describe recent developme

Contents:

Preface; Contents; Chapter 1 Introduction and basic concepts; 1.1 Examples of wave motion; 1.1.1 Sound; 1.1.2 Gravity water waves (linear theory); 1.1.3 Plane electromagnetic wave in vacuum; 1.2 Dispersion; 1.2.1 Evolution of a wave pulse propagating on a shallow water surface (linear theory); 1.2.2 Evolution of electromagnetic pulse in dispersive linear medium; 1.3 Nonlinearity; 1.3.1 Rarefaction wave and dam problem; 1.3.2 Hopf equation and wave breaking; 1.3.3 Simple wave and piston problem; 1.3.4 Characteristics and Riemann invariants

1.3.5 Hodograph transform and a general case of a polytropic gas flow1.4 Nonlinearity and viscosity: Burgers equation; 1.4.1 Derivation of the Burgers equation; 1.4.2 Formation of a shock wave; 1.5 Nonlinearity and dispersion: Korteweg-de Vries equation; 1.5.1 Derivation of the Korteweg-de Vries equation; 1.5.2 Cnoidal wave and soliton; 1.6 Nonlinearity and dispersion: nonlinear Schrödinger equation; 1.6.1 Derivation of the nonlinear Schrödinger equation in nonlinear optics; 1.6.2 Dark soliton solution of the defocusing NLS equation

1.6.3 Modulational instability and the soliton solution of the focusing nonlinear Schrödinger equationBibliographic remarks; Exercises on Chapter 1; Chapter 2 Nonlinear wave equations in physics; 2.1 Korteweg-de Vries equation and modified Korteweg-de Vries equation; 2.1.1 Ion-acoustic nonlinear waves in plasma; 2.1.2 Shallow water waves with account of surface tension; 2.1.3 Waves in nonlinear lattices; 2.2 Nonlinear Schrödinger equation; 2.2.1 Waves in a chain of interacting pendulums; 2.2.2 Modulation of a wave on the shallow water surface; 2.2.3 Deep water surface waves

2.3 Derivative nonlinear Schrödinger equation2.3.1 Nonlinear Alfvén wave; 2.3.2 Dispersionless limit, characteristic speeds and Riemann invariants; 2.3.3 Modulational instability of a plane wave; 2.3.4 Small amplitude nonlinear waves on the constant background; 2.4 Spin waves in magnetic materials; 2.5 Self-induced transparency; 2.6 Stimulated Raman scattering; Bibliographic remarks; Exercises on Chapter 2; Chapter 3 Whitham theory of modulations; 3.1 The general idea; 3.2 Modulation of a linear wave; 3.3 Modulation of a wave-train solution of the nonlinear Klein-Gordon equation

3.4 A variational approach to the modulation theory3.5 Whitham modulational equations for a wave-train solution of the KdV equation; Bibliographic remarks; Exercises on Chapter 3; Chapter 4 Complete integrability of nonlinear wave equations; 4.1 Complete integrability of the KdV equation; 4.1.1 Lamé equation; 4.1.2 Band structure of the Lamé equation; 4.1.3 KdV equation as a compatibility condition of two linear equations; 4.1.4 The KdV hierarchy and the conservation laws; 4.1.5 KdV equation as a Hamiltonian system; 4.1.6 Periodic solution of the KdV equation

4.1.7 Periodic solution of the KdV hierarchy

Notes:

Description based upon print version of record.

Includes bibliographical references (p. 373-380) and index.

ISBN:

9789812792259

9812792252