The Complex Wkb Method For Nonlinear Equations I

When this book was first published (in Russian), it proved to become the fountainhead of a major stream of important papers in mathematics, physics and even chemistry; indeed, it formed the basis of new methodology and opened new directions for research. The present English edition includes new examples of applications to physics, hitherto unpublished or available only in Russian. Its central mathematical idea is to use topological methods to analyze isotropic invariant manifolds in order to obtain asymptotic series of eigenvalues and eigenvectors for the multi-dimensional Schrödinger equation, and also to take into account the so-called tunnel effects. Finite-dimensional linear theory is reviewed in detail. Infinite-dimensional linear theory and its applications to quantum statistics and quantum field theory, as well as the nonlinear theory (involving instantons), will be considered in a second volume.

In this paper we shall discuss the construction of formal short-wave asymp totic solutions of problems of mathematical physics. The topic is very broad. It can somewhat conveniently be divided into three parts: 1. Finding the short-wave asymptotics of a rather narrow class of problems, which admit a solution in an explicit form, via formulas that represent this solution. 2. Finding formal asymptotic solutions of equations that describe wave processes by basing them on some ansatz or other. We explain what 2 means. Giving an ansatz is knowing how to give a formula for the desired asymptotic solution in the form of a series or some expression containing a series, where the analytic nature of the terms of these series is indicated up to functions and coefficients that are undetermined at the first stage of consideration. The second stage is to determine these functions and coefficients using a direct substitution of the ansatz in the equation, the boundary conditions and the initial conditions. Sometimes it is necessary to use different ansiitze in different domains, and in the overlapping parts of these domains the formal asymptotic solutions must be asymptotically equivalent (the method of matched asymptotic expansions). The basis for success in the search for formal asymptotic solutions is a suitable choice of ansiitze. The study of the asymptotics of explicit solutions of special model problems allows us to "surmise" what the correct ansiitze are for the general solution.

The monograph is devoted mainly to the analytical study of the differential, pseudo-differential and stochastic evolution equations describing the transition probabilities of various Markov processes. These include (i) diffusions (in particular,degenerate diffusions), (ii) more general jump-diffusions, especially stable jump-diffusions driven by stable Lévy processes, (iii) complex stochastic Schrödinger equations which correspond to models of quantum open systems. The main results of the book concern the existence, two-sided estimates, path integral representation, and small time and semiclassical asymptotics for the Green functions (or fundamental solutions) of these equations, which represent the transition probability densities of the corresponding random process. The boundary value problem for Hamiltonian systems and some spectral asymptotics ar also discussed. Readers should have an elementary knowledge of probability, complex and functional analysis, and calculus.

This book addresses the peculiarities of nonlinear wave propagation in waveguides and explains how the stratification depends on the waveguide and confinement. An example of this is an optical fibre that does not allow light to pass through a density jump. The book also discusses propagation in the nonlinear regime, which is characterized by a specific waveform and amplitude, to demonstrate so-called solitonic behaviour. In this case, a wave may be strongly localized, and propagates with a weak change in shape. In the waveguide case there are additional contributions of dispersion originating from boundary or asymptotic conditions. Offering concrete guidance on solving application problems, this essentially (more than twice) expanded second edition includes various aspects of guided propagation of nonlinear waves as well as new topics like solitonic behaviour of one-mode and multi-mode excitation and propagation and plasma waveguides, propagation peculiarities of electromagnetic waves in metamaterials, new types of dispersion, dissipation, electromagnetic waveguides, planetary waves and plasma waves interaction.The key feature of the solitonic behaviour is based on Coupled KdV and Coupled NS systems. The systems are derived in this book and solved numerically with the proof of stability and convergence. The domain wall dynamics of ferromagnetic microwaveguides and Bloch waves in nano-waveguides are also included with some problems of magnetic momentum and charge transport.

The 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.

Localized Dynamics of Thin-Walled Shells focuses on localized vibrations and waves in thin-walled structures with variable geometrical and physical characteristics. It emphasizes novel asymptotic methods for solving boundary-value problems for dynamic equations in the shell theory, in the form of functions which are highly localized near both fixed and moving lines/points on the shell surface. Features First-of-its-kind work, synthesizing knowledge of the localization of vibrations and waves in thin-walled shells with a mathematical tool to study them Suitable for researchers working on the dynamics of thin shells and also as supplementary reading for undergraduates studying asymptotic methods Offers detailed analysis of wave processes in shells with varying geometric and physical parameters