mKdV方程的可积离散化

mKdV方程作为描述非谐调晶格中声波的一个模型方程,可用来研究尘埃等离子体中的尘埃孤波,非线性光学中的波动问题等,因此对mKdV方程的解的研究具有重要的实际意义。主要研究了mKdV方程的可积离散化。首先利用适当的变换将mKdV方程转化为连续意义下的双线性导数方程,接着运用双曲算子将所得的mKdV方程的双线性导数方程进行离散化,得到离散的mKdV方程的双线性导数方程。然后通过Hirota小参数扰动方法,对所得的离散的mKdV方程的双线性导数方程进行求解,可求出其单孤子解和二孤子解,并给出这个双线性导数方程的解的一般形式,进而证明了它的可积性。最后应用Matlab软件画出了离散的mKdV方程的双线性导数方程的二孤子解的图形。

Integrable discretization of mKdV equation

mKdV equation is a model equation to describe acoustic in the non-harmonized lattice,and can be used to study dust solitary waves in dust plasma and fluctuations in nonlinear optics,etc.Therefore,it is important to study solutions of rnKdV equation in physical background and practical significance.This paper mainly studies the integrable discretization of mKdV equation.Firstly,the mKdV equation is transformed into continuous bilinear derivative by using of an appropriate transformatior.Then,the continuous bilinear derivative equation of mKdV equation is discretized by the hyperbolic operator,bilinear derivative equation of discrete mKdV equation is got.Next,applying the Hirota small parameter perturbation method,the resulting bilinear derivative equation of discrete mKdV equation is solved,its singlesoliton solutions and two-soliton solutions can be obtained,the general form of the solution for bilinear derivative equation is given,thus its integrability is proved.Finally,the graph of the two-soliton solution of bilinear derivative equation for discrete mKdV equation is given by Matlab.