Description
Book SynopsisThis 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.
Table of ContentsCompact Traveling Waves, Peakons, Cuspons, Solitons, Kinks and Periodic Solutions of Several Third Order Nonlinear PDE, including Camassa-Holm, Korteweg-De Vries, Burger's Equations and Their Modifications; Cellular Neural Network Realization; Fluxon and Breathon Solutions of the Sin-Gordon Equation and Their Interaction; Stability of Periodic Traveling Wave Solutions for Some Classes of KdV Type Equations; Interaction of Peakon Type Solutions of the Camassa-Holm Equation; Classical and Continuous Weak Solutions of the Cauchy Problem for the Hunter-Saxton Equation, Peakon Type Solutions; Weak Continuous Solutions for the Scalar Conservation Law and Existence Of δ-Shocks; Logarithmic Singularities and Microlocal Approach in Studying the Propagation of Nonlinear Waves.
