| As far as we know,many nonlinear phenomena in real life are often described by nonlinear evolution equations,including biology,chemistry,fluid dynamics,physics,economics,mechanics,engineering and many other aspects.With the growth of science and technology,linear models are no longer sufficient to reflect the flexible laws of the impersonal society.In order to better inquire about these physical phenomena,researchers often resort to the solutions of these nonlinear evolution equations.The discovery of a series of nonlinear wave equations in the history of mathematics is in the 20 th century,which is an significant research topic in the field of integrable systems.This has also attracted many scholars to participate in the research in this area,and solve a large number of nonlinear wave equations.The research of soliton theory plays an essential and important role in many fields.Therefore,the solution of the soliton equation is a very significant research theme in doctrine and practice.The main work of this paper is as followsFirst,we solve the local fractional ordinary differential equations through the extended operator method.We list the definitions of local fractional derivatives.Secondly,we prove some properties of a definition operator.Third,we use the extended operator method to obtain the operator representation solution of the local fractional ordinary differential equation.Finally,with the help of the obtained operator representation form the solution and the fractional complex transformation,we solve a local fractional ordinary differential equation,The outcome shows that the extended operator method can be used to solve other local fractional evolution equations.Secondly,We construct Darboux transformation of the(2+1)-dimensional fractional Schr(?)dinger-Maxwell-Bloch equation(SMB)equation,By using Darboux transformation method,the single soliton solution and exact solutions are obtained from the "seed" solution.Firstly,We extend the(2+1)-dimensional SMB equation to the fractional form,Then the Darboux transformation of the(2+1)-dimensional fractional SMB equation is constructed.Finally,some exact solutions of the(2+1)-dimensional fractional SMB equation are given.At last,we use exp-function method and reduction transformations to obtain a pair of rouge wave solutions,which belongs to the DS equation.At start,transform the DS equation into two equations,which are easy to solve.One is the deformed nonlinear Schrodinger(NLS)equation,and the other is a polynomial equation.Second,based on the known solution of the known deformed NLS equation,the rouge wave solution of DS equation is obtained by exp-function method.In the end,we give some space structure and dynamic evolution diagrams,we have obtained rogue wave solutions. |
