简介
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 loarithmic singularities. As it concerns numerical methods in the case of periodic waves the authors apply Cellular Neural Network (CNN) approach.
目录
Preface p. v
Compact travelling waves and peakon solutions p. 1
Introduction and main results p. 1
Proof of Theorem 1.1 p. 4
Proof of Theorem 1.2 p. 8
Generalization of the results p. 11
CNN realization p. 13
Existence and profiles of the travelling waves p. 19
Introduction p. 19
Camassa-Holm type equations p. 19
Generalization of the Burger's equation p. 22
Two component Camassa-Holm system p. 26
Travelling wave solutions of special type to third order nonlinear PDE p. 31
Introduction and main results p. 31
Proof of Theorems 3.1 and 3.2 p. 38
Elliptic functions and applications p. 43
CNN realization p. 60
Cauchy problem for the sin-Gordon equation p. 64
Stability of periodic travelling wave solutions for KDV type equations p. 69
Introduction and main results p. 69
Some results from the theory of elliptic functions p. 75
Interaction of two peakons satisfying the Camassa-Holm equation p. 85
Introduction. Construction of two peakon solutions p. 85
Interaction of two peakons p. 90
Method of the characteristics applied to the Hunter-Saxton equation p. 97
Classical solutions of the Cauchy problem p. 97
Weak solutions of the Hunter-Saxton equation p. 102
Integrable multicomponent Generalizations of Camassa-Holm equation p. 111
Peakon solutions of Hunter-Saxton equation p. 111
Explicit formulas for the peakon-type solutions p. 119
驴-shocks for quasilinear hyperbolic systems in R2 p. 123
Existence of 驴-shocks p. 123
Weak continuous solutions for the scalar conservation laws p. 135
Microlocal approach in studying the propagation of nonlinear waves p. 149
Propagation of jump discontinuities in $$$ p. 149
Creation of logarithmic singularities in $$$ p. 150
Bibliography p. 159
Index p. 167
Compact travelling waves and peakon solutions p. 1
Introduction and main results p. 1
Proof of Theorem 1.1 p. 4
Proof of Theorem 1.2 p. 8
Generalization of the results p. 11
CNN realization p. 13
Existence and profiles of the travelling waves p. 19
Introduction p. 19
Camassa-Holm type equations p. 19
Generalization of the Burger's equation p. 22
Two component Camassa-Holm system p. 26
Travelling wave solutions of special type to third order nonlinear PDE p. 31
Introduction and main results p. 31
Proof of Theorems 3.1 and 3.2 p. 38
Elliptic functions and applications p. 43
CNN realization p. 60
Cauchy problem for the sin-Gordon equation p. 64
Stability of periodic travelling wave solutions for KDV type equations p. 69
Introduction and main results p. 69
Some results from the theory of elliptic functions p. 75
Interaction of two peakons satisfying the Camassa-Holm equation p. 85
Introduction. Construction of two peakon solutions p. 85
Interaction of two peakons p. 90
Method of the characteristics applied to the Hunter-Saxton equation p. 97
Classical solutions of the Cauchy problem p. 97
Weak solutions of the Hunter-Saxton equation p. 102
Integrable multicomponent Generalizations of Camassa-Holm equation p. 111
Peakon solutions of Hunter-Saxton equation p. 111
Explicit formulas for the peakon-type solutions p. 119
驴-shocks for quasilinear hyperbolic systems in R2 p. 123
Existence of 驴-shocks p. 123
Weak continuous solutions for the scalar conservation laws p. 135
Microlocal approach in studying the propagation of nonlinear waves p. 149
Propagation of jump discontinuities in $$$ p. 149
Creation of logarithmic singularities in $$$ p. 150
Bibliography p. 159
Index p. 167
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