Bifurcation and Solitary Waves of the Combined KdV and KdV Equation

Bifurcation, bistability and solitary waves of the combined KdV and mKdV equation are investigatedsystematically. At first, bifurcation and bistability are analyzed by selecting an integral constant as the bifurcationparameter. Then, different conditions expressed in terms of the bifurcation parameter are obtained for the existence ofbreather-like, algebraic, pulse-like solitary waves, and shock waves. All types of the solitary wave and shock wave solutionsare given by direct integration. Finally, an approximate analytic method by employing the interpolation polynomials iscomplete and the theoretical methods are the simplest hitherto.