Analytical Solutions Of A Class Of Fractional Optical Soliton Dynamics Partial Differential Equations

Nonlinear fractional partial differential equation is an important mathematical model widely used in the field of mathematical physics and engineering.Studying the analytical solutions of nonlinear fractional partial differential equations is very helpful for understanding complex engineering system problems.As there is no unified definition of fractional calculus,finding effective methods to solve fractional differential equations is currently an important and meaningful topic.In the thesis,the F-expansion method and bifurcation theory of dynamical system are utilized to extract the analytical solutions of a class of fractional optical soliton dynamics partial differential equations.Firstly,by using F-expansion method,the space-time fractional Sasa-Satsuma equation and space-time fractional perturbed Gerdjikov-Ivanov equation based on two different fractional derivatives are transformed into ordinary differential equation combined with fractional complex transformation,and then the appropriate auxiliary function is selected and the analytical solutions of the two fractional partial differential equations are obtained with the help of Maple software.The two-dimensional and threedimensional diagrams of the partial solutions are drawn by numerical simulation,and the effect of the value of fractional order on the dynamic behavior of soliton solution is discussed.Secondly,by using the bifurcation theory of dynamical systems,the two-dimensional singular traveling wave dynamical system and its regular system corresponding to the space-time fractional complex Ginzburg-Landau equation with nonlinear Power law term are obtained combined with fractional complex transformation,and then the phase diagram of the system under different parameters is drawn,the singular point and bifurcation orbits of the equation in the phase diagram are analyzed,and the exact traveling wave solutions of the corresponding orbits are obtained by integrating different orbits.Finally,by comparing with the solutions obtained in the previous literature,the solutions obtained in this thesis are the new form of solutions.The analytical solutions obtained in this thesis include bright solitons,dark solitons,kink solitons,Compacton solutions,singular periodic solutions and darksingular soliton solutions and so on.The solution obtained in the thesis,from the result,enriches the solution space of fractional optical soliton dynamics partial differential equations and has certain physical meanings.From the process,it reflects the simplicity,efficiency and universality of the F-expansion method and bifurcation theory of dynamical system.