Book Review: Nonlinear nonlocal equations in the theory of waves

Abstract

The nonlinear evolution equations describe various processes andphenomena in physics, chemistry, population dynamics, etc. A largenumber of publications has been devoted to the nonlinear equationsin relation to their numerous applications to science and technology.\parThe present book investigates the nonlocal nonlinear equations appearingin the theory of waves. The methods developed in it are applicableto a wide class of conservative and dissipative nonlinear equations.A major part of the book deals with the problem of breaking and decayingof solutions of nonlinear evolution equations in finite time. Theauthors also investigate existence in the large of classical solutionsand analyze the asymptotics of solutions to various nonlinear evolutionequations. The book contains 10 chapters.\par Chapter 1 is introductory.The authors introduce the concepts of conservation laws, solitarywaves, wave peaking, etc. Chapters 2 and 3 are dealing with the Cauchyproblem and the periodic problem for the Whitham equation, respectively.Various theorems on local existence of classical solutions are presentedin Chapter 2. Results on global existence and smoothing of solutionsare given. Chapter 4 investigates the system of equations of surfacewaves. The existence results are analogous to the results in theprevious chapters. Breaking of solutions is also investigated andtwo-sided estimates for the time of breaking are given. Chapter 5is devoted to generalized solutions of the Cauchy problem for thenonlinear nonlocal equations and systems. Chapter 6-9 are dealingwith the asymptotics, as t \to \infty, of solutions to the Cauchyproblem for generalized Kolmogorov-Petrovski-Piskunov equation, Whithamequation, nonlinear nonlocal Schrödinger equation and system ofequations of surface waves, respectively. The authors find asymptoticformulas, as t \to \infty, for the solutions of the Cauchy problemfor the above equations via incorporating of a special diagram techniqueof the perturbation theory. Chapter 10 is devoted to the step-decayingproblem for the Korteweg-de Vries-Burgers equation.\par The bookis a good exposition of important results. Many interesting examplesare considered. The style of the book is systematic and readable.