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

Abstract

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 also obtained 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.