Nonlinear Waves: An Introduction

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 δ-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.

Riemann-Hilbert Problems with canonical normalization and families of commuting operators

“…These relations are satisfied identically as a consequence of the NLEE (30). The third one also vanishes since tr [L, M ] = 0.…”

“…The development of the soliton theory revealed an important class of NLEE (nonlinear evolution equations) that describe special types of wave-wave interactions [1,29,16,4,23,19,30] which play important role in various fields in physics.…”

Camassa–Holm Cuspons, Solitons and Their Interactions via the Dressing Method

“…In addition the Camassa-Holm equation also allows for the existence of breaking wave solutions, which are realised as solutions which remain bounded but whose gradient becomes unbounded in a finite time, cf. [3,4,9,10,7,53]. The presence of both peaked and breaking wave solutions for the system (CH) ensures the Camassa-Holm equation is a highly interesting physical model.…”

Numerical Investigation Of Nonlinear Waves Connected To Blood Flow In An Elastic Tube With Variable Radius

“…Nonlinear phenomena are usual for physics and especially for fluid mechanics [1] - [10]. One of the most interesting nonlinear phenomena are the nonlinear waves propagating in various media [11] - [17] and especially in fluids [18] - [21]. In this paper we shall discuss nonlinear waves connected to blood flow in large arteries [22] - [24].…”