论文标题
实现Korteweg-de Vries方程的反向散射变换方法
Realization of the inverse scattering transform method for the Korteweg-de Vries equation
论文作者
论文摘要
提出了一种用于实现Korteweg-De Vries方程的反散射转换方法的方法。它基于作者最近获得的JOST解决方案的分析表示和转换运算符的整体内核。表示形式具有功能序列的形式，其中第一个系数在解决直接散射和反向散射问题方面起着至关重要的作用。直接散射问题减少了在简单的复发整合过程之后对许多系数的计算，并通过众所周知的公式对散射数据进行后验计算。反向散射问题还原为线性代数方程的系统，从该系统中，溶液向量的第一个分量导致电势的回收率。我们证明了有限截面方法对线性代数方程系统的适用性，并讨论了提出方法的数值方面。给出了数值示例，其中揭示了该方法的准确性和速度。
A method for practical realization of the inverse scattering transform method for the Korteweg-de Vries equation is proposed. It is based on analytical representations for Jost solutions and for integral kernels of transformation operators obtained recently by the authors. The representations have the form of functional series in which the first coefficient plays a crucial role both in solving the direct scattering and the inverse scattering problems. The direct scattering problem reduces to computation of a number of the coefficients following a simple recurrent integration procedure with a posterior calculation of scattering data by well known formulas. The inverse scattering problem reduces to a system of linear algebraic equations from which the first component of the solution vector leads to the recovery of the potential. We prove the applicability of the finite section method to the system of linear algebraic equations and discuss numerical aspects of the proposed method. Numerical examples are given, which reveal the accuracy and speed of the method.
