New generalized hyperbolic functions and auto-Bäcklund transformation to find new exact solutions of the (2+1)-dimensional NNV equation

In the present Letter, we first time define new functions (called generalized hyperbolic functions) and devise new kinds of transformation (called generalized hyperbolic function transformation) to construct new exact solutions of nonlinear partial differential equations. Based on the generalized hyperbolic function transformation and the auto-Bäcklund transformation of the (2+1)-dimensional Nizhnik-Novikov-Veselov (NNV) equations, we obtain abundant families of new exact solutions of the NNV equations and analyze the properties of them by taking different parameter values of the generalized hyperbolic functions and different regions of the independent variables in these solutions by their figures. As a result, we find that these parameter values and the region size of the independent variables affect some solution structure. Therefore, we may regulate these parameter values and the region size to control the shape and number of solitons on the basis of our different requirements by means of computer simulation. These solutions may be useful to explain some physical phenomena.