Interaction of Solitons as Compound Structures in the Gardner Model

Abstract

Analysis of the modernized approximate description of the interaction of solitary waves as particles allows one to find a new representation of the known exact N-soliton solutions which have the form of a superposition of solitons whose parameters vary relatively slowly. The approximate solutions have the same form as the exact ones, differing from the latter by only the accuracy of representation of the spatio-temporal dependence of the solitary-wave parameters. The approximate description exhibits efficiently the details of the collision, which significantly complements the general picture of soliton interaction within the framework of particular models. These statements are discussed using the example of soliton interaction within the framework of the Gardner model. The specific features of the interaction in this case are caused by the compound structure of solitons, which allows one to analyze them as a superposition of kinks (field steps) of different polarity. We develop a clear and rigorous picture of soliton interaction, which enables us to find previously unknown solutions. In particular, such solutions can have the form of envelope kinks changing the structure of the initial wave field during propagation along infinite sequences (chains) of solitons.