Inverse Scattering Transform for the Multi‐Component Nonlinear Schrödinger Equation with Nonzero Boundary Conditions
Inverse Scattering Transform for the Multi‐Component Nonlinear Schrödinger Equation with Nonzero Boundary Conditions
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DOI：
10.1111/j.1467-9590.2010.00504.x
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发表时间：
2011-04
影响因子：
2.7
通讯作者：
B. Prinari;G. Biondini;A. D. Trubatch
中科院分区：
文献类型：
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作者：
B. Prinari;G. Biondini;A. D. Trubatch
通过调整和推广Beals、Deift和Tomei在N波相互作用方程的逆散射变换（IST）中所使用的方法，推导了具有任意数量分量和空间无限大处非零边界条件的散焦矢量非线性薛定谔方程（NLS）的逆散射变换（IST）。具体而言，一套完整的部分亚纯本征函数是从一个家庭的解析形式，为此目的而构建。与标量和双分量散焦NLS一样，正问题和反问题在两层亏格为零的黎曼曲面上进行表述，然后通过适当的一致化变量将其转换到复平面上。逆问题被表述为具有指定极点、跳跃和对称条件的矩阵Riemann-Hilbert问题。与传统的IST公式不同，首先定义了散射参数的复值的解析形式和本征函数，并将其后验扩展到连续谱。
The Inverse Scattering Transform (IST) for the defocusing vector nonlinear Schrödinger equations (NLS), with an arbitrary number of components and nonvanishing boundary conditions at space infinities, is formulated by adapting and generalizing the approach used by Beals, Deift, and Tomei in the development of the IST for the N‐wave interaction equations. Specifically, a complete set of sectionally meromorphic eigenfunctions is obtained from a family of analytic forms that are constructed for this purpose. As in the scalar and two‐component defocusing NLS, the direct and inverse problems are formulated on a two‐sheeted, genus‐zero Riemann surface, which is then transformed into the complex plane by means of an appropriate uniformization variable. The inverse problem is formulated as a matrix Riemann‐Hilbert problem with prescribed poles, jumps, and symmetry conditions. In contrast to traditional formulations of the IST, the analytic forms and eigenfunctions are first defined for complex values of the scattering parameter, and extended to the continuous spectrum a posteriori.