Multisolitonic solutions from a Bäcklund transformation for a parametric coupled Korteweg-de Vries system

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 $\epsilon$- deformed system we get the infinite sequence of conserved quantities for the parametric coupled system. We also obtain a Bäcklund 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.

In this second version of 23 pages we also considered static solutions which play the role of a background for the system. We also discussed about the regularity of the solutions. We added some figures illustrating the solutions, including the one solitonic solution interacting with the static-background one. We modify the abstract and conclusions in view of these improvements

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• Full hamiltonian structure for a parametric coupled Korteweg-de Vries system