Nonlocal symmetries and explicit solutions for the Gardner equation

From the known Lax pair of the Gardner equation, the Lie symmetry group method is successfully applied to find exact invariant solutions for the Gardner equation with nonlocal symmetries by introducing suitable auxiliary dependent variables. Based on the prolonged system, the explicit analytic solutions related to the Jacobi elliptic functions are derived. Figures illustrate the interaction between the cnoidal waves and a kink wave.