Exact Solutions To Several Nonlinear Evolution Equations

To find exact solutions to soliton equations is an important aspect in soliton theory. It is of important significance in using Hirota's bilinear method, Backlund transformation, nonlinear superposition formula and so on to seek the soliton solutions or other form solutions of several classical evolution equations, such as KdV equation, mKdV equation, Schrodinger equation and the equations generalized from them. This paper mainly stud-ies those four equations:generalized nonlinear Schrodinger equation with the variable coefficient(vc-NLS), the mKdV6 equation, the Schrodinger equation with derivatives and the AKNS hierarchy's first and second equa-tions.The specific work of this dissertation consists as follows:In part one, we first introduce the background of the Schrodinger equa-tion, then we briefly show the definition and elementary properties of the bilinear operators. In the next step, we deduce the bilinear forms of the vc-NLS equation in detail. Then we use the Hirota's bilinear method to get Hirota dark N-soliton solution. In the end, a few figures of solutions under several different cases are shown when aleatoric constants and variables are given exact values.In part two, we first introduce the origin of the mKdV6 equation, then the Hirota form and the N-soliton solution of this equation are considered in the meanwhile. Next, we discuss the Backlund transformation under the rational transform, at the same time some results are derived from it. And the nonlinear superposition formula is obtained in final. The solution of the nonlinear evolution equation in the form of 9 func-tion is mainly studied in part three. The first step is to introduce the defi-nition of the 9 function and some important formulas developed from the combination of Hirota operators and theθfunction. Secondly, the exact solutions of three equations using the method above are obtained, includ-ing the Schrodinger equation with derivatives and the first and the second equations of AKNS hierarchy. During the research process, we come to understand the different treatments of bilinear derivative equations in find-ing the soliton solution and the solution in 9 form. It is shown that there are many powerful tools in searching for exact solutions of soliton equa-tions, and simultaneously we come to recognize the pros and cons of those methods.