This collection consists of four papers in different areas of mathematical physics united by the intrinsic coherence of the asymptotic methods used. The papers describe both the known results and most recent achievements, as well as new concepts and ideas in mathematical analysis of quantum and wave problems. In the introductory paper 'Quantization and Intrinsic Dynamics', a relationship between quantization of symplectic manifolds and nonlinear wave equations is described and discussed from the viewpoint of the weak asymptotics method (asymptotics in distributions) and the semiclassical approximation method. It also explains a hidden dynamic geometry that arises when using these methods.Three other papers discuss applications of asymptotic methods to the construction of wave-type solutions of nonlinear PDE's, to the theory of semiclassical approximation (in particular, the Whitham method) for nonlinear second-order ordinary differential equations, and to the study of the Schrodinger type equations whose potential wells are sufficiently shallow that the discrete spectrum contains precisely one point. All the papers contain detailed references and are oriented not only to specialists in asymptotic methods but also to a wider audience of researchers and graduate students working in partial differential equations and mathematical physics.
Quantization and intrinsic dynamics by M. Karasev Weak asymptotics method and interaction of nonlinear waves by V. G. Danilov, G. A. Omelyanov, and V. M. Shelkovich Global asymptotics and quantization rules for nonlinear differential equations by M. V. Karasev and A. V. Pereskokov Asymptotics of eigenfunctions in shallow potential wells and related problems by P. Zhevandrov and A. Merzon.