The Study Of Solving Methods For Nonlinear Evolution Equations

With the developing of the Nonlinear Sicence, lots of Nonlinear Evolution Equations (NLEEs) are found, and they are playing important roles in many different physics fields. As a leading subject and hot interest in nonlinear science, study on the solution method of nonlinear evolution equations has become more and more challenging.For the complexity of NLEEs, there is no a general solving method for NLEEs. Although exact solutions for a lot of NLEEs have been studied, the solving process has each skill respectively, thus there still have many NLEEs whose exact solutions are unable to be obtained. For this reason, sometimes we don't solve the equations, but research about properties for their solutions accoding to equation itself directly.There are some methods for solving NLEEs, such as homegeneous balancing method,Tanh function method,Backlund transformation,Inverse scattering method,Darboux transformation,Hirota bilinear methods similarity reduction eta. Base on Division Theorem and Hilbert-Nullstellensatz Theorem, Z. S. Feng proposed a new approach for studing the compound Burgers-KdV equation in 2002, which is currently called the first integral method and becoming an effective method for solving some NLEEs.Based on the theory of nonlinear evolution equations, and according to the above theory and methods, and with the aid of computer symbolic computing system, five aspects work are completed in this paper.Firstly, with the aid of symbolic computation system Maple and byusing the hyperbolic tangent function (?)tanh(ξ)ⁱ, 36 groups of traveling wave solutions for nonlinear Aceive-dissipative dispersive equation u_t + uu_x + u_xx + pu_xxx + u_xxxx = 0 are obtained. The method can be also used to solve other nonlinear wave equations.Secondly, by using superposition method, the exact solutions of Aceive-dissipative dispersive equation u_t + uu_x + u_xx + pu_xxx + u_xxxx = 0are obtained;Thirdly, by using the Painleve test, its application in Hirota bilinear is studied;Forth, applying the first integral method, we obtain exact solutions of (2+1) Burgers equation;Fifth, this article proves some sufficient conditions using elementary integrations and gives some examples for those specific Riccati equations.The studies are of more profoundly theoretical significance and important application value. The paper "Travelling-wave solution of nonlinear Aceive-dissipative dispersive equation" which based on the first part is accepted by "Journal of Jiamusi University" and the paper "Elementary Solutions for the Riccati Equation "which based on the fifth part is accepted by "Journal of Xuzhou Normal University".