Analytical Solutions For Two New Nonlinear Integrable Equations

Generally speaking, it is very difficult to solve the analytical solutions of nonlinear partial differential equations. In this paper, we establish some relationship between two new nonlinear integrable equations and two-well known equations by some transformation. With the aid of symbolic computation system Mathematica, we have done the following research:(1) In the second chapter, under reciprocal transformation, new nonlinear integrable equation with arbitrary constant ? transformed into well-known KdV equation, and new nonlinear integrable equation with arbitrary constant ? and ? transformed into Gardner's equation. The obtained parametric representations solutions of two new nonlinear integrable equations include periodic wave solution, rational solution, smooth soliton solution, hyperbolic type kink and rational type kink, and their figures are plotted.(2) The aim of the third chapter, is that we look for the N-soliton solutions of the new nonlinear integrable equation with arbitrary constant ?by Darboux transformation combined with reciprocal transformation.Parametric representations of multi-soliton solutions for the new nonlinear integrable equation are obtained, then we study in detail the interactions of multi-soliton solutions.