The single solitary wave with double kinks of the combined KdV equation and its stability

Abstract

Based on the ideas of the hyperbola function expansion method, we obtained some analytical solutions of the combined KdV-mKdV (cKdV) equations by introducing new expansion functions.One of the single soliton solutions has the kink-antikink structure, and this solution reduces to the kink-like solution and the bell-like solution under different limitations. Theoretical analysis shows that the cKdV equation has both propagated-type and non-propagated-type solitary wave solutions. We also investigated the stability of the single solitary wave solution with double kinks numerically. The results indicate that the solution may be stable or unstable, depending on different sets of parameters.