Study On Exact Solutions And Integrability Of Some Nonlinear Evolution Equations

In the real world,many phenomena can be generalized as mathematical models of nonlinear evolution equations.People begin to have a wide interest in solving nonlinear evolution equations.Moreover,nonlinear evolution equations have rich backgrounds and h ave been widely used in biology,mathematics,physics and many other fields.But for the exact solutions,there is no general method.Therefore,we need to find an approach to solve this problem.Many scientists studied the characteristics of nonlinear evolution equations,and finally Lie symmetry analysis method was proposed and achieved a great development in both theorical and practical fields.Based on this,the analytical solutions and properties of two kinds of fractional order partial differential equations are studied by using the classical Lie symmetry method.The main work of this paper includes the following aspects:1.On the premise of Riemann Liouville fractional differential operator,the time fractional CRWP equation is analyzed by using Lie symmetry method,and thus the classical Lie symmetry method is extended from integer differential equation to fractional differential equation.By using the symbolic calculation software Maple,the infinitesimal generator of the time fractional CRWP equation is obtained,and the similarity reduction of the equation is further obtained,then the equation is transformed into a fractional ordinary differential equation containing the Erdelyi-Kober differential operator.At the same time,the power series solutions of the equation is obtained by using the theorem of power series.Then by choosing different parameters of the power series,we analyze its convergence and the influence on power series solutions.Finally,a new conservation vector of the time-fractional CRWP equation is constructed,and the convergence of the power series’ solution is proved.2.The space-time fractional Burgers equation is a physical process which describes the weak nonlinear sound wave propagating in the gas pipeline along one direction.First,Lie symmetry method of fractional partial differential equations is applied to the space-time fractional partial differential equations.Second,we apply the definition of the Lie symmetry to calculate the infinitesimal generator of the equation,then the equation is transformed into a fractional ordinary differential equation.Finally,by using a new conservation theorem and Lagrange form,a new conservation vector of the conservation law of the space-time fractional Burgers equation is constructed.