In the present book the reader will find a review of methods for constructing a certain class of asymptotic solutions, which we call self-stabilizing solutions. This class includes solitons, kinks, traveling waves, etc. It can be said that either the solutions from this class or their derivatives are localized in the neighborhood of a certain curve or surface. For the present edition, the book published in Moscow by the Nauka publishing house in 1987, was almost completely revised, essentially up-dated, and shows our present understanding of the problems considered. The new results, obtained by the authors after the Russian edition was published, are referred to in footnotes. As before, the book can be divided into two parts: the methods for constructing asymptotic solutions ( Chapters I-V) and the application of these methods to some concrete problems (Chapters VI-VII). In Appendix a method for justification some asymptotic solutions is discussed briefly. The final formulas for the asymptotic solutions are given in the form of theorems. These theorems are unusual in form, since they present the results of calculations. The authors hope that the book will be useful to specialists both in differential equations and in the mathematical modeling of physical and chemical processes. The authors express their gratitude to Professor M. Hazewinkel for his attention to this work and his support.

Klappentext

This book presents an up-to-date review of methods for constructing a specific class of asymptotic solutions, called self-stabilizing solutions. This class includes solitons, kinks, travelling waves, etc. Either the solutions for this class or their derivatives are localized in the neighbourhood of a certain curve of surface. br/ The book can be divided into two parts: Chapters I-V contain the methods for constructing asymptotic solutions, and in Chapters VI-VII these are applied to some concrete problems. The Appendix briefly discusses a method for the justification of some asymptotic solutions. br/ emAudience:/em This book will be of interest to specialists both in differential equations and in the mathematical modelling of physical and chemical processes.

Inhalt
I. Properties of Exact Solutions of Nondegenerate and Degenerate Ordinary Differential Equations.- 1.1. Standard equations.- 1.2. Examples.- II. Direct Methods for Constructing Exact Solutions of Semilinear Parabolic Equations.- 2.1. Preliminary notes.- 2.2. Representation of self-similar solutions in terms of rational functions.- 2.3. Construction of exact one-phase and two-phase solutions.- 2.4. Formulas for solutions of semilinear parabolic equations with common cubic nonlinearity.- 2.5. Relation between the number of phases in the solution and the degree of nonlinearity.- 2.6. Asymptotics of wave creation for the KPP-Fisher equation.- III. Singularities of Nonsmooth Solutions to Quasilinear Parabolic and Hyperbolic Equations.- 3.1. Main definitions.- 3.2. Asymptotic solutions bounded as ? ? 0.- 3.3. Asymptotic solutions unbounded as ? ? 0.- 3.4. The structure of singularities of solutions to quasilinear parabolic equations near the boundary of the solution support.- 3.5. The structure of singularities of nonsmooth self-similar solutions to quasilinear hyperbolic equations.- IV. Wave Asymptotic Solutions of Degenerate Semilinear Parabolic and Hyperbolic Equations.- 4.1. Self-stabilizing asymptotic solutions.- 4.2. Construction of nonsmooth asymptotic solutions. Derivation of basic equations.- 4.3. Global localized solutions and regularization of ill-posed problems.- 4.4. Asymptotic behavior of localized solutions to equations with variable coefficients.- 4.5. Heat wave propagation in nonlinear media. Asymptotic solutions to hyperbolic heat (diffusion) equation.- 4.6. Localized solutions in the multidimensional case.- V. Finite Asymptotic Solutions of Degenerate Equations.- 5.1. An example of constructing an asymptotic solution.- 5.2. Asymptotic solutions in theone-dimensional case.- 5.3. Asymptotic finite solutions of degenerate quasilinear parabolic equations with small diffusion.- 5.4. Relation between approximate solutions of quasilinear parabolic and parabolic equations.- VI. Models for Mass Transfer Processes.- 6.1. Nonstationary models of mass transfer.- 6.2. Asymptotic solution to the kinetics equation of nonequilibrium molecular processes with external diffusion effects.- 6.3. The simplest one-dimensional model.- VII. The Flow around a Plate.- 7.1. Introduction.- 7.2. Uniformly suitable asymptotic solution to the problem about the flow of low-viscous liquid around a semi-infinite thin plate.- 7.3. Asymptotic behavior of the laminar flow around a plate with small periodic irregularities.- 7.4. Critical amplitude and vortices in the flow around a plate with small periodic irregularities.- References.- Appendix. Justification of Asymptotic Solutions.- 1. One-dimensional scalar case.- 3. Zeldovich waves.