Residual symmetries and soliton-cnoidal wave interaction solutions for the negative-order Korteweg-de Vries equation

The residual symmetry is derived for the negative-order Korteweg-de Vries equation from the truncated Painlevé expansion. This nonlocal symmetry is transformed into the Lie point symmetry and the finite symmetry transformation is presented. The multiple residual symmetries are constructed and localized by introducing new auxiliary variables, and then nth Bäcklund transformation in terms of determinant is provided. With the help of the consistent tanh expansion (CTE) method, the explicit soliton-cnoidal wave interaction solutions are obtained from the last consistent differential equation.