The inverse scattering transform for the focusing nonlinear Schrodinger equation is presented for a general class of initial conditions whose asymptotic behavior at infinity consists of counterpropagating waves. The formulation takes into account the branched nature of the two asymptotic eigenvalues of the associated scattering problem. The Jost eigenfunctions and scattering coefficients are defined explicitly as single-valued functions on the complex plane with jump discontinuities along certain branch cuts. The analyticity properties, symmetries, discrete spectrum, asymptotics and behavior at the branch points are discussed explicitly. The inverse problem is formulated as a matrix Riemann-Hilbert problem with poles. Reductions to all cases previously discussed in the literature are explicitly discussed. The scattering data associated to a few special cases consisting of physically relevant Riemann problems are explicitly computed.

中文翻译：

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具有反向传播流的聚焦非线性薛定谔方程的逆散射变换

聚焦非线性薛定谔方程的逆散射变换是针对一类一般的初始条件提出的，这些初始条件在无穷远处的渐近行为由反向传播波组成。该公式考虑了相关散射问题的两个渐近特征值的分支性质。Jost 特征函数和散射系数被明确定义为复平面上的单值函数，沿着某些分支切割具有跳跃不连续性。明确讨论了分支点处的解析性、对称性、离散谱、渐近性和行为。逆问题被表述为具有极点的矩阵 Riemann-Hilbert 问题。之前在文献中讨论过的所有案例的简化都得到了明确的讨论。