Soliton interactions (for the Korteweg-deVries equation): a new perspective

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Abstract
It can be shown that the N-soliton solution of the Korteweg-deVries equation can be decomposed into N separate solitons (cf. Gardner et al. (1974), Calogero and Degasperis and Yoneyama (1982)). However, it is not immediately clear from the form of their solutions how the separate solitons relate directly to the single soliton solution. Here the two-soliton case is considered and a decomposition is sought which can be clearly related to the single soliton solution. Although it appears that there is a family of such decompositions it is shown that only one of these is correct. Although this decomposition is equivalent to the decomposition given previously by Gardner et al., Calogero and Degasperis and Yoneyama, the form given here is different. It is suggested that the form of solution produced here is a more appropriate representation of the solution since it is clear how it relates directly to the single soliton solution and it is easy, through this form, to analyse the interaction of the two solutions.

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