Residual symmetry, interaction solutions and consistent tanh expansion solvability for the third-order Burgers equation

The residual symmetries for the third-order Burgers equation are obtained with the truncated Painlevé method. The nonlocal symmetries can be localized to the Lie point symmetries by introducing auxiliary dependent variables and the corresponding finite transformations are computed directly. New exact solutions of the third-order Burgers equation is also proved to have the consistent tanh expansion form. New exact interaction excitations such as soliton-cnoidal wave solutions and soliton-periodic wave solutions are given out analytically and graphically.