Geometric asymptotics for nonlinear PDE. I. Transl. from the Russian manuscript by D. Chibisov (Q2758007)

It is impossible to describe strongly nonlinear systems in any approximation of the method of local linearization. Therefore, until recently strong nonlinearity seemed to be an insurmountable obstacle for fruitful investigation of physical systems. A class of nonlinear systems, which allow such investigation, was discovered by M. D. Kruskal and N. Zabusky. They have a partial solution in the form of solitary waves, the solitons. A nonlinear generalization of the Fourier method, inverse scattering transform, has appeared. Within the framework of the perturbation approach integrable systems may appear as approximations of systems that are nonintegrable but close to integrable ones. NEWLINENEWLINENEWLINEHowever, this theory has a rather narrow range of applicability. So, the development of an asymptotic theory of nonlinear waves with localized fast variations became desirable. The relatively simple new asymptotic method proposed by \textit{V. P. Maslov} and \textit{G. A. Omel'yanov} in the monograph under review is closely related to geometric asymptotics (WKBJ approach) for linear differential equations. It may be considered as nonlinear generalization of WKBJ. The main idea is formulated by the authors as follows: the geometric object is considered in the configuration space, and all constructions are of a local nature. This conception guides one to the global description.NEWLINENEWLINENEWLINEIn the monograph, apart from methodology, solutions of many difficult problems are given. It is very appealing that almost each concept is introduced and explained through a number of well-selected, easy-to-understand examples, originated from problems of hydrodynamics, nonlinear optics, plasma physics, mechanics of solids, theory of phase transition.NEWLINENEWLINENEWLINEThis well-written book is a reader-friendly and good-organized monograph in the field of nonlinear science. It can be highly recommended for experts in PDE, applied mathematics and various areas of physics.