Equations of Mathematical Physics KdV Equation Camassa-Holm Type Equations Burger's Equation Hunter-Saxton Equation Conservation Laws Equations Traveling Wave Solutions Nonlinear PDE Periodic Solutions of Traveling Type for mKdV Equations Generalized Cauchy Problem Riemann Problem CNN Approach

Issue Date:

22-Nov-2010

Publisher:

University Press "Paisii Hilendarski", Plovdiv

Abstract:

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, conservation laws
equations 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, we 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),
existence and uniqueness results, etc. 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 -shocks are also considered. As it concerns numerical methods we
apply the CNN approach.
The book is addressed to a broader audience including graduate students,
Ph.D. students, mathematicians, physicist, engineers and specialists in the domain
of PDE.