Systematic Methods To Nonlinear Evolution Equations And Their Exact Explicit Solutions

With the development of science and technology, there are many nonlinear problems in natural and social areas, which arouses much concern. These problems are usually characterized by nonlinear evolution partial differential equations (NLEPDEs). So how to construct exact solutions of the associated nonlinear equations plays an important role in understanding the nonlinear problems. Upon now, there are several available methods in solving the NLEPDEs, for example, the inverse scattering transform, the Hirota method, the Backlund transformation method and the Homogeneous method.Chapter1: the developement of the theory of soliton is presented .The famous KdV equation is introduced in detail, which plays an important role in the theory of nonlinear equations. And the problem of the interaction between solitons is also studied, which shows that the traveling solitons keep steady after collision.Chapter2, :the inverse scattering transform is introduced. Its aim is to make the problem of solving a nonlinear evolution equation into the problem of solving three linear equations. Based on this method, the KdV equation on the initial condition is solved . And one-soliton, two-soliton and N-soliton solutions are obtained.Chapter3: the method of Backlund transformation is introduced. Its aim is to build the connection between the solutions of two different equation, or the connection between the solutions of one same equation. And some exact explicit solutions of Burgers equation and Sine-Gordon equation are obtained through separately constructing their Backlund transformations.Chapter4: the traveling wave method is illustrated by solving the KdV equation and Sine-Gordon equation. Two kinds of traveling wave solutions are obtained, namely the periodic solutions and solitary solutions. And their geometrical features are simply studied.Chapter5: the recently developed method of hyperbolic tangent function expansion is extended and new function transformation is applied to construct some new solitary solutions of KdV equation and Klein-Gordon equation And the Jacobielliptic function expansion method, which is advanced in 2001,and the extended method of doubly Jacobi function expansion are used to construct the exact solutions of a kind of nonlinear evolution equations. Thus many new periodic solutions are obtained and the periodic solutions will degenerate into solitary solutions on the limit condition, which shows this method is more powerful. In the part of discussion, the suitability of the Jacobi elliptic function expansion method is also studied by proposing the 'rank'. And we firstly point out that when the 'ranks' of every term of the nonlinear evolution equation are simultaneously even or odd, the method can be used to solve the equation. And the newly developed method of Lame function method in solving the multi-order approximate equations of nonlinear evolution equations is also discussed. And following equations are solved in this method: nonlinear Schr dinger, BBM equation, Zakharov equation, KP equation, Boussinesq equation and cubic nonlinear Schr dinger.