Study Of Systematic Methods To Nonlinear Evolution Equations

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. In this paper, there are several available methods in solving the NLEPDEs, for example, the Homogeneous balance method, the hyperbolic tangent function expansion method, the function transformation method and the Jacobi elliptic function expansion method.Chapter1: firstly, the developement of the theory of soliton is presented The famous KdV equation is introduced detailly , 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 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 wave solutions. And their geometrical features are simply studied.Chapter3: the homogeneous balance method is introduced. By using of the homogeneous balance method, six exact explicit solutions of KdV equation are obtained: the exact balance solution, the solitary wave solution, the rational solution, the mixed solution of the multinomial form and the exponential function, the mixed solution of the multinomial form and the sine-cosine function. As another example of the improved homogeneous balance method, we would like to show the solutions of two-dimension dispersive long-wave equations also can be found by the improved homogeneous balance method.Chapter4: the hyperbolic tangent function expansion method is introduced. Its aim is to make the traveling wave solution of nonlinear evolution equation into the form solution of the hyperbolic tangent function. And the recently developed method of hyperbolic tangent function expansion, as well as its extended hyperbolic functionexpansion, is introduced. Based on these two methods, some solitary wave solutions of KdV equation, nonlinear Klein-Gordon equation and combined KdV equation are obtained.Chapter5: The Jacobi elliptic function expansion method is introduced. And we can also extend this method to a more powerful new method of doubly Jacobi elliptic function expansion. A big patch of exact explicit solutions is obtained by application of the method, which includes the periodic solutions and the solitary wave solutions. In the part of discussion, the suitability of the Jacobi elliptic function expansion method is also studied by proposing the "rank", and we point out that when the "ranks" of every term of the nonlinear evolution equation is completely even or odd, the method can be used to solve the equation. And the newly developed method of the application of Lam6 function in solving the multi-order approximate equations of nonlinear evolution equations is also discussed. Following equations are solved by this method: nonlinear Schrddinger equation, BBM equation, Zakharov equation, etc. And from the definition of Legendre elliptic integration and Jacobi elliptic function, new transformation are obtained and applied to construct the exact solutions of nonlinear evolution equations. These new analytical solutions such as periodic solutions and soliton solutions are deprived of many nonlinear evolution equations.Chapter6: the function transformation method is introduced. Its aim is that the form solution of nonlinear evolution equation is assumed as the function expansion form. By constructing the coefficients of the function expansion form, we can get the function expansion form solutions of nonlinear evolution equation. In this paper, many kinds of function transformation is introduced to solve nonlinear evolution equations, and we can get the soliton-like solution of the generalized KdV equation with variable coefficient, new traveling wave solutions and the solitary wave solutions of Fisher equation and two dimension Burgers-KdV equation, explicit exact solitary wave solutions of the nonlinear derivative Schrddinger equation with fifth-order stronger nonlinear term, new solitary wave solutions and new periodic solutions with sine-cosine function of nonlinear Klein-Gordon equation.