On discontinuous periodic solution and discontinuous solitary wave of two-dimension shallow water equation

In this paper we discuss discontinuous periodic solution and discontinuous solitary wave of the shallow water model of geophysical fluid dynamics. When we consider the properties of trajectory near non-equilibrium point, i.e. singular point, we find that if we introduce the concept of generalized solution (piecewise smoothing continuous solution), then the system will produce discontinues periodic solution and the condition of discontinuous periodic solution can he obtained. When the system is degenerated, we find that the discontinuous solitary wave is existent in the system. In this paper we consider a series of problems and obtain analytic expression of discontinuous solution. This result is compared with squall line in the atmosphere, and both of them have many things in common.