Study On The Exact Solutions Of Two Integrable Equations

The main object of this paper is the integrable evolution equation.The main research contents include the application of the Riemann-Hilbert method in solving the integrable equation,using the Hirota bilinear method to solve the N-soliton solution,lumps,breathers and interaction solutions of the equation.In the first part,we use the Riemann Hilbert method to solve the soliton solution of the generalized nonlinear Schrodinger(GNLS)equation,which consists of the following three steps.First,we analyze the spectrum of the integrable equation's Lax pair.Second,we establish the Riemann Hilbert problem of the equation.Last,we consider the Riemann Hilbert problem corresponding to reflectionless case and give the expression of general N-soliton solutions of the equation.In the second part,we find the N-soliton solution of a(3+1)-dimensional generalized KP equation by the Hirota bilinear method.Then,we obtain the T-order breathers of the equation,and combine the long-wave limit method to give the M-order lumps.Resorting to the extended homoclinic test technique,we obtain the breather-kink solutions for the equation.Last,the interaction solution composed of the K-soliton solution,breathers and lumps for the(3+1)-dimensional generalized KP equation is constructed.