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The space-time ray method. Linear and nonlinear waves. (Prostranstvenno-vremennoj luchevoj metod. Linejnye i nelinejnye volny.) (Russian) Zbl 0678.35002

Leningrad: Izdatel’stvo Leningradskogo Universiteta. 272 p. R. 2.50 (1985).
This book is devoted to a systematic exposition of the STRM (space-time ray method) of constructing asymptotic solutions to certain wave problems. Chapter 1, “STRM for the wave equation”, may be considered as an introduction; here for the wave equation with a variable velocity the notions of the eikonal equation and transport equations are introduced, methods of their solution are presented, a link between nonstationary, quasistationary and stationary asymptotics is discussed; moreover a history of the STRM is described. In Chapter 2, “STRM in vector wave problems”, the method developed is applied to the Maxwell system and to systems in variational form, and a link between STRM and the Whitham average Lagrangian method is discussed. In Chapter 3, “STR-description of wave processes of different physical nature”, the complex STRM is presented and it is applied to an examination of surface scalar waves; moreover by means of real STRM asymptotic formula for nonstationary normal waves in a thin curved waveguide with the variable section, for interior waves in the ocean, for surface waves on a heavy liquid, and for reflected ST-waves from the moving surface, are derived; finally, the asymptotic expansion near ST-caustics is derived. In Chapter 4, “Construction of ST-asymptotics in non-stationary problems”, a link between stationary and nonstationary asymptotics is discussed and analogues of ST-formulae for a large class of nonstationary problems (including theory of waves in a medium with dispersion) are derived. Chapters 5 and 6 are devoted to nonlinear problems. In Chapter 5, “Small nonlinear perturbation of one-frequency oscillatory systems”, two-scaled asymptotic expansions for ordinary differential equations (such as the van der Pol equation) are derived. In Chapter 6, “Some asymptotic approaches in theory of progressing waves in nonlinear media”, STRM is generalized to waves in a medium with a small nonlinearity and small dispersion. In particular quasiperiodic and perturbed soliton solutions are investigated; the nonlinear Klein-Fock-Gordon equation, generalized perturbed Korteweg-de Vries equation and nonlinear Helmholtz equation are considered as examples.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35C20 Asymptotic expansions of solutions to PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
35B50 Maximum principles in context of PDEs
35Lxx Hyperbolic equations and hyperbolic systems
58J45 Hyperbolic equations on manifolds
35B20 Perturbations in context of PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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