Conservation laws for shallow water waves on a sloping beach

Shallow water waves are governed by a pair of nonlinear partial differential equations. We transfer the associated homogeneous and nonhomogeneous systems (corresponding to constant and sloping depth, respectively) to the hodograph plane, where we find all the nonsimple wave solutions and construct infinitely many polynomial conversation laws. We also establish correspondence between conservation laws and hodograph solutions as well as Bäcklund transformations by using the linear nature of the problems on the hodograph plane.