Exact Solutions For Two Kinds Of Nonlinear Equations

With the development of science and technology, more and more naturalphenomena and social problems can be analyzed by nonlinear models. Thesemake nonlinear problems become one of the hot reseach areas. Most of thesenonlinear problems can be described by nonlinear equations. So it is importantand of practical significance to solve the nonlinear equations.In this paper, KdV6equation is studied by the multi-linear variable separa-tion approach, the second type of Riccati equation expansion method, the Exp)expansion method and (G/G)expansion method. The Extended Painleve′ex-pansion is used to examine the Painleve′properties of KdV6equation. Some im-portant nonstandard truncation solutions are obtained. At last, we use (G/G)expansion method to get the exact traveling wave solutions for two models ofphase transitions driven by configurational forces. The main contents of this pa-per are as follows:Chapter one is an introduction. The background of topics, the research ofcurrent KdV6equation and two models of phase transitions driven by configura-tional forces are briefly reviewed.In chapter two the multi-linear variable separation approach are introduced.The basic steps of the multi-linear variable separation method for solving partialdifferential equations, and the use of this method for solving the KdV6equationare given in detail. Two particular solutions of KdV6equation are obtained by using this method. One of them contains an arbitrary function. Some pictures ofthe solution are plotted by choosing different arbitrary functions and parametervalues using Maple software.The method of traveling wave reduction is discussed in chapter three. Threedifferent methods based on the traveling wave reduction are proposed. Whenthe reduction of the nonlinear equations after traveling wave reduction, an ordi-nary differential equation can be obtained, Different treatments are adopted bythese three methods respectively. The second type of Riccati equation and (G/G)function method use the specific function expansion for solving nonlinear equa-tions, while the Exp)function method use the truncated expansion for solvingnonlinear equations. Finally we use the second type of Riccati equation, the Exp-function method and the (G/G))function method to solve the KdV6equation andapply the (G/G)expansion method to solve the one-dimensional phase transitionmodel.Chapter four the Extended Painleve′expansion is introduced. The ExtendPainleve′expansion is first analyzed. Then using this method to check the Painleve′properties of the KdV6equation and to prove that it has Painleve′test. In the endof this chapter, the Extended Painleve′expansion method is used to solve non-standard truncated solutions of the KdV6equation for N=3and N=4.The five chapter is the summary and prospect.