扩充的含零点的黎曼问题方法

扩充的含零点的黎曼问题方法 AN EXTENDED RIEMANN PROBLEM WITH ZEROS 黄念宁 Huang Nianning first-author 对应于反散射法,通常的黎曼问题方法中,零点限制在复λ的上半平面。本文讨论具有任意位置的零点的黎曼问题,并用它来求解非线性薜定谔方程。导出了基本方程而不引用反散射法的任何知识。当零点限制在上半平面时,它直接给出反散射法的Zakharov-Shabat方程。此外将给出新的解,可以利用代数技巧直接作出验证,给出了解的正规性条件,并证实新解中有正规解。 Instead of the usual Riemann problem with zeros in the upper half plane of complex λ which corresponds to the inverse scattering method, an extended one with arbitrary zeros is discussed for solving the nonlinear Schrodinger equetion. Fundamental equations are obtained from the extended Riemann problem with zeros without knowledge of the inverse scattering method. They have the same form with the Zakharov—Shabat equations of the inverse scattering method in the case of λ in the upper half plane. In other cases, they give new solutions which can be also directly verified to satisfying the nonlinear Schrodinger equation. Regularity conditions of the solutions are given and one of the new regular solutions is given as an example. 黎曼问题 孤子解 反散射方法 Riemann problem soliton solution inverse scattering method 国家科学基金 1988-01-01 2021-04-01 1