Residual symmetry, nth Bäcklund transformation, and soliton‐cnoidal wave interaction solution for the combined modified KdV–negative‐order modified KdV equation

Abstract

This paper is concerned with the nth Bäcklund 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 Painlevé 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.