Small-Dispersion Limit of the Korteweg-deVries Equations

摘要

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 exp 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. (ERA citation 08:011042)