A (2+1)-Dimensional Displacement Shallow Water Wave System

A new (2+1)-dimensional shallow water wave system, the (2+1)-dimensional displacement shallow water wave system (2DDSWWS), is constructed by applying variational principle of the analytic mechanics under the Lagrange coordinates. The general travelling wave solution is expressed as an elliptic integral. A special case is explicitly expressed by the Jacobi elliptic function which is a generalization of the solitary wave solution. Compared with some traditional (2+1)-dimensional shallow water wave systems such as the Kadomtsev--Petviashvili (KP) description under the Euler coordinates, the 2DDSWWS has some its own advantages. In addition, the KP equation can also be derived from the 2DDSWWS under the weak two-dimensional long-wave approximation.