Residual Symmetry of the Alice-Bob Modified Korteweg-de Vries Equation**Supported by the National Natural Science Foundation of China under Grant Nos. 11705077 and 11775104

Abstract

Starting from the truncated Painlevé expansion, the residual symmetry of the Alice-Bob modified Korteweg-de Vries (AB-mKdV) equation is derived. The residual symmetry is localized and the AB-mKdV equation is transformed into an enlarged system by introducing one new variable. Based on Lie’s first theorem, the finite transformation is obtained from the localized residual symmetry. Further, considering the linear superposition of multiple residual symmetries gives rises to N-th Bäcklund transformation in the form of the determinant. Moreover, the Ps Td (the shifted parity and delayed time reversal) symmetric exact solutions (including invariant solution, breaking solution and breaking interaction solution) of AB-mKdV equation are presented and two classes of interaction solutions are depicted by using the particular functions with numerical simulation.