反散射变换在Kundu-Eckhaus方程中的应用

Application of Inverse Scattering Transform to Kundu-Eckhaus Equation

由于非线性模型的解可以反映很多数学物理现象,故求解非线性模型的解具有重要意义.反散射变换作为求解非线性可积模型的方法之一,主要步骤是构造其对应方程的Lax对Riemann-Hilbert问题,然后反过来求解Riemann-Hilbert问题的解析解,进而得到方程所对应的解.主要利用反散射变换研究了在零边界条件下的局部Kundu-Eckhaus(KE)方程的孤子解,通过Riemann-Hilbert问题的解研究了N个简单极点情况下的精确孤子解公式,并进行数值模拟,直观地给出了所得到的孤子解.

Because the solution of the nonlinear model can reflect many mathematical physics phenomena,solving the solution of the nonlinear model is of great significance.As one of the methods to the nonlinear integrable model,the main step of the inverse scattering transformation is to construct the Lax pair and Riemann-Hilbert problem of the corresponding equation,and then the analytical solution of the Riemann-Hilbert problem is solved in turn,the corresponding solution of the equation is got.In this thesis,the soliton solutions of the local Kundu-Eckhaus(KE)equation with zero boundary conditions are studied by using the inverse scattering transformation method.The exact soliton solutions formula for the case of N simple poles are studied by using the solutions of the Riemann-Hilbert problem.In addition,numerical simulations is performed and the soliton solution intuitively is given.