Abstract
In this paper, we present a generalized unified method for finding multi-wave solutions of nonlinear evolution equations via the (2+1)-dimensional Nizhnik-Novikov-Veselov equations with variable coefficients (vary with time). Multi-auxiliary equations have been introduced in this method to obtain not only multi-soliton solutions but also multi-periodic or multi-elliptic solutions. Compared with the Hirota's method and the inverse scattering method, the proposed method gives more general exact multi-wave solutions without much extra effort. To give more physical insight to the obtained solutions, we present graphically their representative structures by setting the arbitrary functions in the solutions as specific functions. It is shown that rogue waves are generated in the solutions of the velocity components in an incompressible fluid which they are enveloped by the characteristic curves. Furthermore, we found multi-elliptic waves highly dispersed far from the core of waves.