Soliton equations: admitted solutions and invariances via B\"acklund transformations
Sandra Carillo, Cornelia Schiebold
Nonlinear Sciences, Exactly Solvable and Integrable Systems, Exactly Solvable and Integrable Systems (nlin.SI), Mathematical Physics (math-ph)
A couple of applications of B\"acklund transformations in the study of nonlinear evolution equations is here given. Specifically, we are concerned about third order nonlinear evolution equations. Our attention is focussed on one side, on an invariance admitted by the interacting soliton equation and, on the other one, on the construction of solutions. Indeed, via B\"acklund transformations, a B\"acklund chart, connecting Abelian as well as non Abelian equations can be constructed. The importance of such a net of links is twofold since it indicates invariances as well as allows to construct solutions admitted by the nonlinear evolution equations it relates. The present study refers to third order nonlinear evolution equations of KdV type. On the basis of the Abelian wide B\"acklund chart which connects various different third order nonlinear evolution equations an invariance admitted by the int sol. KdV equation is obtained and an explicit solution is constructed. Then, the corresponding non-Abelian B\"acklund chart, shows how to construct matrix solutions of the mKdV equations: some recently obtained solutions are reconsidered.