LS解法和Fisher方程行波系统的定性分析

提出了求解非线性发展方程的新方法——LS解法.LS解法是基于(G’/G)展开法和扩展的双曲正切函数展开法.并引入了Poincar定性理论的思想,然后以Fisher方程为例进行了试验.通过定性分析首先获得了Fisher方程行波系统积分曲线的性质,然后解得了Fisher方程作为耗散系统时单调减少的波前解和作为扩张系统时单调递增的波前解.一些试验结果与Ablowitz所得结果一致.也得到了Fisher方程作为扩张系统时的新结果.LS解法是在定性理论指导下,在已获知解曲线性质的情况下进行精确求解的,求解目标明确.LS解法揭示了线性系统也可以用作辅助方程来求解非线性系统.

LS method and qualitative analysis of traveling wave solutions of Fisher equation

LS method,a new method for solving nonlinear evolution equations, is proposed. It is based on the (G'/G)-expansion method and the extended hyperbolic tangent function method, and the Poincaré' s qualitative theory is also led in. Then Fisher equation is tested as an example. The properties of integral curves for traveling wave system of Fisher equation are obtained through qualitative analysis, and then a monotonically decreasing wave-front solution of Fisher equation as a dissipative system and a monotonically increasing wave-front solution of Fisher equation as an expansion system are obtained too. Some results agree with that of Ablowitz et al. and some new results for Fisher equation are alsoobtained as an expansion system. The LS method is used to look for the exact solutions under the condition that the property of solution curves have been obtained through the qualitative analysis,and the target is clear. The LS method also reveals that a linear system can also be used as auxiliary equations to solve nonlinear systems.