Study On Exact Solutions Of Some Nonlinear Evolution Equations

Nonlinear evolution equations are important models for describing nonlinear phenomena in the fields of fiber optic communication,nonlinear atmosphere dynamics,plasma physics,biological mathematics,etc.More and more scientists are devoted to studying the exact solutions of these equations to revel the secret behind corresponding natural phenomena.Common methods for solving nonlinear evolution equations l are:Backlund transformation,Hirota bilinear method,Lie symmetry method,Painleve analysis method,Bell polynomial method,auxiliary equation method,and so on.Based on the symbolic computation,we uses the Lie symmetry method and the auxiliary equation method to study the exact solutions and their properties for some nonlinear evolution equations.The main work of this paper has the following aspects:1.Lie symmetry method is used to perform detailed analysis on the modified Zakharov-Kuznetsov(MZK)equation.We have obtained the infinitesimal generators,commutator table of Lie algebra and symmetry group.In addition to the above,optimal system of one-dimensional subalgebras up to conjugacy is derived which is used to construct similarity reduction solutions.These solutions describe the dynamics of nonlinear waves in isothermal multicomponent magnetized plasmas.2.Lie symmetry method is applied to investigate the invariance properties of the time fractional Kolmogorov-Petrovskii-Piskunov(FKPP)equation with Riemann-Liouville derivative.In view of point symmetry,the vector fields for the governing equation are well constructed,And then the equation can be reduced to a fractional ODE with the help of Erdelyi-Kober operator.Moreover,a kind of explicit power series solutions of the equation is derived by virtue of the power series theory.3.Under investigation in this section is a Kundu-Eckhaus equation with variabl coefficients(vKE),which models the propagation of the ultra-short femtosecond pulses in an optical fiber.An intensity-dependent chirp ansatz is applied to reduce the original equation into two coupled amplitude-phase nonlinear equations of the propagating wave.It is shown that the dynamics of field amplitude in this system is governed by a first-order nonlinear ordinary differential equation with a sixth-degree nonlinear term.As a result,a very rich variety of chirped solitons(bright,dark,kink and gray solitons)which has nontrivial phase is derived.Moreover,the constraint conditions that naturally fall out of the solution structure which guarantee the existence of these solitons are also reported.