Maslov V.P., Omel'yanov G.A. Geometric Asymptotics for Nonlinear PDE. I

American Mathematical Society, USA, 2001. — 296 p. — (Translations of Mathematical Monographs 202) — ISBN: 0821821091.The study of asymptotic solutions to nonlinear systems of partial differential equations is a very powerful tool in the analysis of such systems and their applications in physics, mechanics, and engineering. In the present book, the authors propose a new powerful method of asymptotic analysis of solutions, which can be successfully applied in the case of the so-called ""smoothed shock waves"", i.e., nonlinear waves which vary fast in a neighborhood of the front and slowly outside of this neighborhood. The proposed method, based on the study of geometric objects associated to the front, can be viewed as a generalization of the geometric optics (or WKB) method for linear equations. This volume offers to a broad audience a simple and accessible presentation of this new method. The authors present many examples originating from problems of hydrodynamics, nonlinear optics, plasma physics, mechanics of continuum, and theory of phase transitions (problems of free boundary). In the examples, characterized by smoothing of singularities due to dispersion or diffusion, asymptotic solutions in the form of distorted solitons, kinks, breathers, or smoothed shock waves are constructed. By a unified rule, a geometric picture is associated with each physical problem that allows for obtaining tractable asymptotic formulas and provides a geometric interpretation of the physical process. Included are many figures illustrating the various physical effects.Waves in One-dimensional Nonlinear Media
Nonlinear Waves in Multidimensional Media
Asymptotic Solutions of Some Pseudodifferential Equations and Dynamical Systems with Small Dispersion
Problems with a Free Boundary
Multi-phase Asymptotic Solutions
Asymptotics of Stationary Solutions to the Navier-Stokes Equations Describing Stretched Vortices
Appendix: List of Equations