Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV

Abstract

The pole dynamics in the complex plane associated with the two-soliton solution of the Korteweg–de Vries equation are studied in detail. The poles trace smooth curves as time evolves and fall into one of three categories: those which are asymptotic for large negative time to the faster soliton, but for large positive time are asymptotic to the slower soliton, those which follow the opposite pattern of the previous class, and those which are asymptotic for large positive and negative time to the faster soliton. Furthermore, the precise position and time of the interaction is identified. Finally, new examples of finite time blowup of complex-valued solutions of the Korteweg–de Vries equation are found and their asymptotic behaviours at blowup are determined.It is suggested that these findings lend support to the assertions that the leading, slower moving soliton transforms during the interaction into the faster moving soliton, and that a mass–energy transfer takes place between the two solitons.