标题： 非线性薛定谔方程的逆散射变换方法的实现 显示英文标题

标题： Implementation of the inverse scattering transform method for the nonlinear Schrödinger equation

摘要： 我们研究非线性薛定谔方程的初值问题。 逆散射变换方法的应用涉及求解Zakharov-Shabat系统在复势下的直接和逆散射问题。 我们通过使用Zakharov-Shabat系统的Jost解的新级数表示来解决这些问题。 这些表示形式为关于变换后的谱参数的幂级数。 根据这些表示，直接散射问题的求解简化为按照简单的递归积分过程计算级数系数，通过乘以相应的多项式对（级数表示的部分和）计算散射系数，并定位单位圆内的多项式的零点。 逆散射问题的求解简化为求解幂级数系数的线性代数方程组，而势函数则从前几个系数中恢复。 该方程组直接由散射关系得到。 因此，与现有的其他技术不同，该方法不涉及求解Gelfand-Levitan-Marchenko方程或矩阵黎曼-希尔伯特问题。 整体方法导致了一个简单且高效的数值求解非线性薛定谔方程初值问题的算法，这由数值例子加以说明。 显示英文摘要

摘要： We study the initial-value problem for the nonlinear Schr\"odinger equation. Application of the inverse scattering transform method involves solving direct and inverse scattering problems for the Zakharov-Shabat system with complex potentials. We solve these problems by using new series representations for the Jost solutions of the Zakharov-Shabat system. The representations have the form of power series with respect to a transformed spectral parameter. In terms of the representations, solution of the direct scattering problem reduces to computing the series coefficients following a simple recurrent integration procedure, computation of the scattering coefficients by multiplying corresponding pairs of polynomials (partial sums of the series representations) and locating zeros of a polynomial inside the unit disk. Solution of the inverse scattering problem reduces to the solution of a system of linear algebraic equations for the power series coefficients, while the potential is recovered from the first coefficients. The system is obtained directly from the scattering relations. Thus, unlike other existing techniques, the method does not involve solving the Gelfand-Levitan-Marchenko equation or the matrix Riemann-Hilbert problem. The overall approach leads to a simple and efficient algorithm for the numerical solution of the initial-value problem for the nonlinear Schr\"odinger equation, which is illustrated by numerical examples.