1974/03/01 by Robert M. Miura, Martin D. Kruskal · 1 citation
Physics and Astronomy · #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics
paper · doi:10.1137/0126036
crossref issued 1974/03/01 · crossref published 1974/03/01 · crossref published-print 1974/03/01 · openalex publication_date 1974/03/01 · crossref created 2005/02/23 · crossref deposited 2017/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28 · crossref indexed 2026/07/31
The WKB method used in quantum mechanics for solving linear second order ordinary differential equations is generalized to apply to nonlinear partial differential equations. In particular, this nonlinear WKB method, which is similar to the averaging method due to Whitham, is used to study nearly-periodic solutions of the Korteweg–deVries equation when the dispersion parameter is small. The emphasis of this paper is on a detailed analysis of the leading-order problem arising from the application of the nonlinear WKB method. An explicit representation of the leading-order solution is obtained in terms of unknown functions whose qualitative properties are studied. These unknown functions are governed by a first order system of nonlinear partial differential equations which is of hyperbolic type.