Waves and structures in nonlinear nondispersive media. General theory and applications to nonlinear acoustics.
type: Item
instance of: scholarly article
title: Waves and structures in nonlinear nondispersive media. General theory and applications to nonlinear acoustics.
zbMATH Open document ID: 1246.76001
author: S. N. Gurbatov
author: O. V. Rudenko
author: Alexander I. Saichev
published in: Nonlinear Physical Science
publication date: 2011-10-21T00:00:00Z
review text: The book is divided into two parts. The first part (Chapters 1-4) gives a detailed description of basic concepts and methods for analysis of nonlinear wave problems in nondispersive media, while the second part (Chapters 5-11) is devoted to applications in nonlinear acoustics. If possible, the presentation is given on the physical level of rigor, considering only hydrodynamic nonlinear waves in nondispersive media. Chapter 1 examines the solutions of thesimplest nonlinear PDEs of first order: simple wave equations (canonical form of the equation, Riemann solution, construction of the density field, Fourier transforms of density and velocity), line-growth equation of forest-fire propagation, and one-dimensional laws of gravitation (Lagrangian and Eulerian descriptions). Chapter 2 discusses generalized solutions to PDEs of first order (multistream solutions, weak solutions). The main ideas and solution methods for nonlinear PDEs of second order are presented in Chapter 3 with the help of the examples of the Kardar-Parisi-Zhang equation (arising as a regularized version of the line-growth equation, and describing the deposition of semiconductor films) and the Burgers equation (arising as a regularization of first order PDE of nonlinear acoustics). The regularization is understood here as the introduction into the equation of additional terms containing higher order derivatives to prevent the gradient catastrophe. Two sections of Chapter 3 study general solutions of the Burgers equation, demonstrating the effectivity of standard methods for investigation of nonlinear phenomena. Some modifications and generalizations of the Burgers equation admitting general solutions are discussed as model equations of gas dynamics. In Chapter 4, the authors describe some solutions of the Burgers equation for a single-scale field (evolution of one-dimensional signals) with discussion of typical scenarios of solution evolution realized under various initial conditions with special attent…
Mathematics Subject Classification ID: 76-02
Mathematics Subject Classification ID: 76Q05
Mathematics Subject Classification ID: 76L05
Mathematics Subject Classification ID: 76D33
Mathematics Subject Classification ID: 00A79
zbMATH DE Number: 5961790
zbMATH Keywords: line-growth equation
zbMATH Keywords: Kardar-Parisi-Zhang equation
zbMATH Keywords: Burgers equation
zbMATH Keywords: group analysis
zbMATH Keywords: shock wave
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5961790