The small dispersion limit of the korteweg‐de vries equation. ii

Abstract

Abstract In Part I* we have shown, see Theorem 2.10, that as the coefficient of u xxx tends to zero, the solution of the initial value problem for the KdV equation tends to a limit u in the distribution sense. We have expressed u by formula (3.59), where ψ x is the partial derivative with respect to x of the function ψ* defined in Theorem 3.9 as the solution of the variational problem formulated in (2.16), (2.17). ψ* is uniquely characterized by the variational condition (3.34); its partial derivatives satisfy (3.51) and (3.52), where I is the set I o defined in (3.36). In Section 4 we show that for t<t b , I consists of a single interval, and the u satisfies u t — 6 uu x = 0; here t b is the largest time interval in which (12) has a continuous solution. In Section 5 we show that when I consists of a finite number of intervals, u can be described by Whitham's averaged equation or by the multiphased averaged equations of Flaschka, Forest, and McLaughlin. Equation numbers refer to Part I.