Analysis Of The Exact Solution Of Nonlinear Equations And Solution Method

With the development of science and technology and the needs of practical application, people are faced with a large number of non-linear problems, many of which needs to be depicted with nonlinear equation. Therefore, the study of nonlinear equations is very important. Through the long-term efforts of experts and scholars, many effective methods have been raised, such as inverse scattering method, Backlund transformation, Darboux transformation, Hirota bilinear method, Lie method, variable separation method, Painleve expansion and so on. Recently, with the development of computer algebras, some effective and direct methods have been raised. For example the homogeneous balance method, F-expansion method and its extended form, hyperbolic function method and its extension or amendment, trial function method, Jacobi elliptic function method and its extended form, auxiliary equation method, Riccati equation method, projection Riccati equation method, direct reduction method, Sine-Cosine method,the generalized power– exponential function method. Through these methods we can get various exact solutions of nonlinear equations. Another question arises: are the solutions in different mathematics expressions really different in essence to the corresponding nonlinear equation?In this paper, we study the (2+1)-dimensional dispersive long wave equations, and extend analytical solution to the method analysis and systematically summarize and analyze the hyperbolic function method, give the general steps of hyperbolic function method at last.The first part briefly introduces the development backgrounds of nonlinear science.In the second part, we first introduce the general principles and new applications of the homogeneous balance method, then we obtained Backlund transformation of the two-dimensional dispersive long wave equations. Finally, we get four exact solutions of the equations by Backlund transformations.In the third part, firstly we introduce the general steps of hyperbolic function method, and get six groups solutions of (2+1)-dimensional dispersive long wave equations by this method, then we give four groups solutions by Sirendaoreji in [23] using auxiliary equation method, through identical deformation we find that they are equivalent to the four solutions we have got. Finally we find these solutions are equivalent through two different methods. So we systematically analyze the expansion of the hyperbolic function method proposed in the literature by Yan, Dai, Xie, Huang, Shang. However, the results show that these methods are mathematically different, but essentially equivalent.