Residual symmetry, nth Backlund transformation, and soliton-cnoidal wave interaction solution for the combined modified KdV-negative-order modified KdV equation

摘要

This paper is concerned with the nth Backlund transformation (BT) related to multiple residual symmetries and soliton-cnoidal wave interaction solution for the combined modified KdV-negative-order modified KdV (mKdV-nmKdV) equation. The residual symmetry derived from the truncated Painleve expansion can be extended to the multiple residual symmetries, which can be localized to Lie point symmetries by prolonging the combined mKdV-nmKdV equation to a larger system. The corresponding finite symmetry transformation, ie, nth BT, is presented in determinant form. As a result, new multiple singular soliton solutions can be obtained from known ones. We prove that the combined mKdV-nmKdV equation is integrable, possessing the second-order Lax pair and consistent Riccati expansion (CRE) property. Furthermore, we derive the exact soliton and soliton-cnoidal wave interaction solutions by applying the nonauto-BT obtained from the CRE method.