The Korteweg-de Vries Two-Soliton Solution as Interacting Two Single Solitons

Abstract

The exact two-soliton solution of the Korteweg-de Vries (KdV) equation u is decomposed into a simple sum u, + U2, which can be regarded as attractive scattering of two single solitons according to this decomposition. Each Ui satisfies corresponding Interacting KdV equation which is physically natural extension of the original KdV equation. Nonlinear phenomena have been studied exten­ sively for these two decades since the discovery of soliton by Zabusky and Kruskal, ') but still no definite interpretation seems to exist about the interaction of solitons, that is, whether it is at­ tractive or repulsive. This is because N -soliton solution U(N) has been studied as a whole and no detailed analysis has been done so far. We can construct a picture that several single solitons initially apart in space come close, inter­ act with each other satisfying certain coupled nonlinear differential equations and eventually become apart again in space without changing their identity. In our general theory, we will introduce some new operators and decompose the exact N -soliton solution U(N) into N parts, and examine each term to see how it behaves and what equation it satisfies_ They interact always attractively according to our decomposition regardless of the height ratios. But in this paper we concentrate our attention on the simplest N = 2 case, which is very interesting and important. We consider here the Korteweg-de Vries (KdV) equation for u(x, t) in the following form: