Details

Autor(en) / Beteiligte

• Maslov, Victor P

Titel

The Complex WKB Method for Nonlinear Equations I : Linear Theory [Elektronische Ressource]

Ist Teil von

• Progress in Physics : 16

Ort / Verlag

Basel : Birkhäuser Basel

Erscheinungsjahr

1994

Link zu anderen Inhalten

• Volltext

Beschreibungen/Notizen

• This book deals with asymptotic solutions of linear and nonlinear equations 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 asymptotics 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 problems of mathematical physics; certain specific formulas were obtained by different methods (V. M. Babich [5 -7], V. P. Lazutkin [76], A. A. Sokolov, 1. M. Ternov [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 procedure for fields with complex phase, similar to the usual one for fields with real phase, will be developed