The Complex Wkb Method for Nonlinear Equations I: Linear Theory

By Victor P. Maslov

Book Overview

This book deals with asymptotic solutions of linear and nonlinear equa- tions which decay as h ---+ 0 outside a neighborhood of certain points, curves and surfaces. Such solutions are almost everywhere well approximated by the functions cp(x) exp{iS(x)/h}, x E 1R3, where S(x) is complex, and ImS(x) o. When the phase S(x) is real (ImS(x) = 0), the method for obtaining asymp- totics of this type is known in quantum mechanics as the WKB-method. We preserve this terminology in the case ImS(x) 0 and develop the method for a wide class of problems in mathematical physics. Asymptotics of this type were constructed recently for many linear prob- lems of mathematical physics; certain specific formulas were obtained by differ- ent methods (V. M. Babich 5 -7], V. P. Lazutkin 76], A. A. Sokolov, 1. M. Ter- nov 113], J. Schwinger 107, 108], E. J. Heller 53], G. A. Hagedorn 50, 51], V. N. Bayer, V. M. Katkov 21], N. A. Chernikov 35] and others). However, a general (Hamiltonian) formalism for obtaining asymptotics of this type is clearly required; this state of affairs is expressed both in recent mathematical and physical literature. For example, the editors of the collected volume 106] write in its preface: "One can hope that in the near future a computational pro- cedure for fields with complex phase, similar to the usual one for fields with real phase, will be developed.

• Format: Paperback
• Language: English
• ISBN: 3034896697
• ISBN13: 9783034896696
• Release Date: October 2012
• Publisher: Birkhauser
• Length: 304 Pages
• Weight: 0.98 lbs.
• Dimensions: 0.7" x 6.1" x 9.2"