Part I of this series appeared in Comm. Pure Appl. Math. 36, 1983, pp. xxx-xxx.
The small dispersion limit of the korteweg-de vries equation. ii†
Article first published online: 18 OCT 2006
Copyright © 1983 Wiley Periodicals, Inc., A Wiley Company
Communications on Pure and Applied Mathematics
Volume 36, Issue 5, pages 571–593, September 1983
How to Cite
Lax, P. D. and Levermore, C. D. (1983), The small dispersion limit of the korteweg-de vries equation. ii. Comm. Pure Appl. Math., 36: 571–593. doi: 10.1002/cpa.3160360503
- Issue published online: 18 OCT 2006
- Article first published online: 18 OCT 2006
- Manuscript Received: JUN 1982
- U.S. Department of Energy. Grant Number: DE-AC02-76ER03077
In Part I* we have shown, see Theorem 2.10, that as the coefficient of uxxx 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 ψx 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 Io defined in (3.36). In Section 4 we show that for t<tb, I consists of a single interval, and the u satisfies ut — 6uux = 0; here tb 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.