Local Fractional Derivative Of Inverse Matrix And Mixed Spectral AKNS Equations

Solving nonlinear evolution equations has been paid attention to by many scholars both in theory and in application.Because of its complexity,it is very difficult to solve nonlinear evolution equations.However,many scholars have established and developed some methods for nonlinear evolution equations through continuous exploration and research.Among them,the method of inverse scattering transform has been in the hot phase and has been widely used.The main steps of inverse scattering transform can be divided into two steps,one is the direct scattering,the other is inverse scattering.We can construct the corresponding spectral problem and its time evolution equation to get eigen function of the spectral problem through the direct scattering analysis on the initial potential,and hence gain the scattering data at the zero time,from which reach the time-dependence of above scattering data.Then we can reconstruct the potential by inverse scattering analysis and obtain soliton solutoins of nonlinear evolution equations by solving the GLM equation in the case of reflectionless potential.This is the famous inverse scattering transform.It is helpful to understand the nonlinear phenomena described by some complex equations by means of inverse scattering transform,and it has certain promotion to academic research in related fields,which is worthy of further study.With the development of fractional calculus and fractal calculus,the derivative of inverse matrix also plays an important role in the field of nonlinear mathematical physics.It has been used to derive some famous nonlinear partial differential equations.The main research work of this paper,on the one hand,we give a formula of the local fractional partial derivative of inverse matrix and applies it to soliton theory,so as to derive the local fractional non-isospectral self-dual Yang-Mills equation and the local fractional principle chiral field equation;On the other hand,a new generalized mixed spectrum AKNS equation is constructed by selecting appropriate spectral parameters,and then the inverse scattering transform method is extended to the derived AKNS equation,and soliton solutions are obtained in the case of reflectionless potential.The specific arrangement is asfollows:Firstly,the background of soliton and inverse scattering transform are introduced briefly,and the development of fractional-order partial differential equations is summarized.Secondly,a formula of the local fractional partial derivative of inverse matrix is given and proved.With the help of the derived formula,the local fractional non-isospectral self-dual Yang-Mills equation and the local fractional principle chiral field equation are derived.Finally,a new generalized mixed spectrum AKNS equation is constructed,and the inverse scattering transform is applied to the AKNS equation to obtain the soliton solutions,and the dynamic properties of some solutions are studied.