Soliton excitations and periodic waves without dispersion relation in shallow water system

Using the nonstandard and standard truncations of a modified Conte’s invariant Painlevé expansion for the dispersive long wave equation system, two types of soliton excitations without any dispersive relations are found. Four types of periodic waves expressed by Jacobi elliptic functions are found by the truncations of a special extended Painlevé expansion. The soliton solutions are special cases of the corresponding two of the given periodic solutions. The dispersion relations of the solutions are crucially dependent on the boundary conditions.