Solving Two Fractional-order Equations By Hirota's Billinear Method And Inverse Scattering Transform

With the development and applications of fractional order calculus equation,kinetics and dynamics of fractional order system has caused the attention of many scholars,so some important analytical method can be in the field of nonlinear partial differential equation of fractional order have better application,and what about the fractional order partial differential equations of fractional order soliton dynamics and integrability,became some notable problems.Hirota's bilinear method and inverse scattering transformation method are famous methods for solving nonlinear partial differential equations.Hirota's bilinear method makes the equation bilinear under proper transformation,and then the exact solution of the equation can be obtained through calculation and derivation.On the one hand,this paper studies how to apply Hirota bilinear method to solve nonlinear fractional partial differential equations,obtain its exact solution,and further study the dynamic properties of fractional solution of n-soliton solutions.On the other hand,the inverse scattering transformation is generalized and applied to solve nonlinear fractional partial differential equations(PDEs)to gain exact solutions.The main work of this paper has:Firstly,a local fractional order KP(LFKP)equation with Lax integrability is derived and solved by the generalized Hirota bilinear method.After the appropriate transformation,the LFKP equation is bilinearized.On the basis of the bilinear form,the n-soliton solution containing the Mittag-Leffler function is obtained.Secondly,by the known spectral problems in equipment spectrum function of local time the development of the fractional order derivative equation,the inverse scattering transformation method is extended to nonlinear partial differential equations,and take the local time fractional order KdV equation as an example,by using inverse scattering transformation,on the basis of the local time fractional order derivative spectrum problem,by generalized inverse scattering transformation,derivation and solved with Lax integrability of fractional order KdV equations,then get the exact solution of formula containing Mittag-Leffler function.finally,in the case of no reflection potential,take the exact solution reduced to fractional n-soliton solution,in addition,in order to have a deeper understanding of fractional-order n-soliton dynamics,the dynamic evolution images of fractional order single-soliton solutions,double-soliton solutions and three-soliton solutions are simulated.The velocity of soliton solution obtained by image research changes with the change of fractional order.