Small-dispersion limit of the Korteweg-deVries equations

Technical Report · Sat May 01 00:00:00 EDT 1982

OSTI ID:6638709

Lax, P D; Levermore, C D

The scattering transform method is used to study the weak limit of solutions to the initial-value problem for the Korteweg-deVries (KdV) equation as the dispersion tends to zero. In that limit the associated Schroedinger operator becomes semiclassical, so the exact scattering data is replaced by its corresponding WKB expressions. Only nonpositive initial data are considered; in that case the limiting reflection coefficient vanishes. The explicit solution of Kay and Moses for the reflectionless inverse transform is then analyzed, and the weak limit, valid for all time, is characterized by a quadratic minimum problem with constraints. With the aid of function theoretical methods, solutions of this minimum problem are constructed in terms of solutions to an initial value problem for some auxiliary functions. The weak limit satisfies the KdV equation with the dispersive term dropped until the finite time at which its derivatives become infinite. Up to that time the weak limit is a strong L/sup 2/ limit. At later times the weak limit is locally described by Whitham's averaged equations or, more generally, by the equations found by Flaschka et al. using multiphase averaging. For large times, behavior of the weak limit is studied directly from the minimum problem.

• Research Organization: New York Univ., NY (USA). Courant Mathematics and Computing Lab.
• DOE Contract Number: AC02-76ER03077
• OSTI ID: 6638709
• Report Number(s): DOE/ER/03077-172; ON: DE83004385
• Country of Publication: United States
• Language: English

Related Subjects

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
INVERSE SCATTERING PROBLEM
KORTEWEG-DE VRIES EQUATION
LIMITING VALUES
ANALYTICAL SOLUTION
DISPERSION RELATIONS
SCATTERING
WKB APPROXIMATION
DIFFERENTIAL EQUATIONS
EQUATIONS
PARTIAL DIFFERENTIAL EQUATIONS
645500* - High Energy Physics- Scattering Theory- (-1987)
658000 - Mathematical Physics- (-1987)