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Nonlinear Waves : An Introduction - Petar Radoev Popivanov

Nonlinear Waves

An Introduction

Hardcover Published: 24th September 2010
ISBN: 9789814322126
Number Of Pages: 168

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This book deals with equations of mathematical physics as the different modifications of the KdV equation, the Camassa-Holm type equations, several modifications of Burger's equation, the Hunter-Saxton equation and others. The equations originate from physics but are proposed here for their investigation via purely mathematical methods in the frames of university courses. More precisely, the authors propose classification theorems for the traveling wave solutions for a sufficiently large class of third order nonlinear PDE when the corresponding profiles develop different kind of singularities (cusps, peaks). The orbital stability of the periodic solutions of traveling type for mKdV equations are also studied. Of great interest too is the interaction of peakon type solutions of the Camassa-Holm equation and the solvability of the classical and generalized Cauchy problem for the Hunter-Saxton equation. The Riemann problem for special systems of conservation laws and the corresponding d-shocks are also considered. At the end of the book the authors study the interaction of two piecewise smooth waves in the case of two space variables and they verify the appearance of logarithmic singularities. As it concerns numerical methods in the case of periodic waves the authors apply Cellular Neural Network (CNN) approach.

Prefacep. v
Compact travelling waves and peakon solutionsp. 1
Introduction and main resultsp. 1
Proof of Theorem 1.1p. 4
Proof of Theorem 1.2p. 8
Generalization of the resultsp. 11
CNN realizationp. 13
Existence and profiles of the travelling wavesp. 19
Introductionp. 19
Camassa-Holm type equationsp. 19
Generalization of the Burger's equationp. 22
Two component Camassa-Holm systemp. 26
Travelling wave solutions of special type to third order nonlinear PDEp. 31
Introduction and main resultsp. 31
Proof of Theorems 3.1 and 3.2p. 38
Elliptic functions and applicationsp. 43
CNN realizationp. 60
Cauchy problem for the sin-Gordon equationp. 64
Stability of periodic travelling wave solutions for KDV type equationsp. 69
Introduction and main resultsp. 69
Some results from the theory of elliptic functionsp. 75
Interaction of two peakons satisfying the Camassa-Holm equationp. 85
Introduction. Construction of two peakon solutionsp. 85
Interaction of two peakonsp. 90
Method of the characteristics applied to the Hunter-Saxton equationp. 97
Classical solutions of the Cauchy problemp. 97
Weak solutions of the Hunter-Saxton equationp. 102
Integrable multicomponent Generalizations of Camassa-Holm equationp. 111
Peakon solutions of Hunter-Saxton equationp. 111
Explicit formulas for the peakon-type solutionsp. 119
¿-shocks for quasilinear hyperbolic systems in R2p. 123
Existence of ¿-shocksp. 123
Weak continuous solutions for the scalar conservation lawsp. 135
Microlocal approach in studying the propagation of nonlinear wavesp. 149
Propagation of jump discontinuities in $$$p. 149
Creation of logarithmic singularities in $$$p. 150
Bibliographyp. 159
Indexp. 167
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9789814322126
ISBN-10: 9814322121
Series: Series on Analysis, Applications and Computation
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 168
Published: 24th September 2010
Country of Publication: SG
Dimensions (cm): 22.86 x 15.24  x 1.78
Weight (kg): 0.45