Multisolitonic solutions from a B\"acklund transformation for a parametric coupled Korteweg-de Vries system

TL;DR

This paper introduces a parametric coupled KdV system, derives a Bäcklund transformation, and generates new multisolitonic and periodic solutions, including solutions on non-trivial static backgrounds.

Contribution

It develops a generalized Gardner transformation and Bäcklund transformation for the parametric coupled KdV system, enabling the construction of new multisolitonic solutions.

Findings

Generated new multisolitonic solutions depending on multiple parameters.

Proved permutability theorem for the Bäcklund transformation.

Found solutions propagating on non-trivial static backgrounds.

Abstract

We introduce a parametric coupled KdV system which contains, for particular values of the parameter, the complex extension of the KdV equation and one of the Hirota-Satsuma integrable systems. We obtain a generalized Gardner transformation and from the associated $\varepsilon$- deformed system we get the infinite sequence of conserved quantities for the parametric coupled system. We also obtain a B\"{a}cklund transformation for the system. We prove the associated permutability theorem corresponding to such transformation and we generate new multi-solitonic and periodic solutions for the system depending on several parameters. We show that for a wide range of the parameters the solutions obtained from the permutability theorem are regular solutions. Finally we found new multisolitonic solutions propagating on a non-trivial regular static background.