A two dimensional distributed soliton solution of the Korteweg-de Vries equation

It is suggested that recent discrete soliton solutions of the Korteweg-de Vries equation in two dimensions can be generalized to solutions which have a continuous variation of parameters. In the particular case of the resonant interaction of solitons, as discussed by Miles, an analytic solution is obtained in which only one parameter is varied. The structure of this solution is examined in detail. Asymptotic solutions for large positive and negative time are constructed as well as solutions at large distance for finite time. The asymptotic analysis is found to be similar to that developed by Lighthill for Burgers' equation. A resonant triad of discrete solitons as described by Miles is shown to develop into a curved soliton. Numerical computations confirm the asymptotic analysis. (author)