The (2+1)-dimensional integrable coupling of KdV equation: Auto-Bäcklund transformation and new non-traveling wave profiles

The (2+1)-dimensional integrable coupling of the KdV equation, which was first presented by Ma and Fussteiner from the celebrated KdV equation using the perturbation method of multiple scales u=u1+εu2, y=εx, is investigated. With the aid of symbolic computation, a new auto-Bäcklund transformation is gained and used to seek new types of non-traveling wave solutions involving an arbitrary function of y. Moreover a non-traveling wave similarity variable transformation reduces this system to a system of non-linear ordinary differential equations with constant coefficient, which is solved to get non-traveling wave Jacobi elliptic function solutions and Weierstrass elliptic function solutions involving an arbitrary smooth function of y. When the arbitrary function are taken as some special functions, these obtained solutions possess abundant structures. The figures corresponding to these solutions are illustrated to show the rules of the wave propagation related to (2+1)-dimensional integrable coupling of the KdV equation.