Inverse Scattering Transform for the Defocusing Manakov System with Non-Parallel Boundary Conditions at Infinity
Inverse Scattering Transform for the Defocusing Manakov System with Non-Parallel Boundary Conditions at Infinity
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DOI：
10.4208/eajam.261021.230122
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发表时间：
2022-06
影响因子：
1.2
通讯作者：
Asela Abeya;G. Biondini;B. Prinari
中科院分区：
文献类型：
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作者：
Asela Abeya;G. Biondini;B. Prinari
.散焦Manakov系统的逆散射变换（IST）是用无限非零边界条件（包括非平行边界条件）开发的，即，渐近极化向量形式主义使用一个均匀化变量映射到一个单一的副本的复平面的频谱平面的两个副本，从而消除平方根分支。“伴随”的拉克斯对也被用来克服一些Jost本征函数的非解析性的问题。反问题是制定一个合适的矩阵黎曼-希尔伯特问题（RHP）。与平行边界条件相比，IST中最显著的区别是散射系数的渐近行为，这影响了RHP中本征函数和分段亚纯矩阵的归一化。当渐近极化矢量不正交时，给出了两种不同的方法将RHP转化为一组线性代数积分方程。然而，当渐近极化矢量正交时，这些方法中只有一种是适用的。最后，证明了在正交和非正交极化矢量的情况下，不存在无反射势，这意味着该问题不存在纯孤子解.
. The inverse scattering transform (IST) for the defocusing Manakov system is developed with non-zero boundary conditions at infinity comprising non-parallel boundary conditions — i.e., asymptotic polarization vectors. The formalism uses a uniformization variable to map two copies of the spectral plane into a single copy of the complex plane, thereby eliminating square root branching. The “adjoint” Lax pair is also used to overcome the problem of non-analyticity of some of the Jost eigenfunctions. The inverse problem is formulated in term of a suitable matrix Riemann-Hilbert problem (RHP). The most significant difference in the IST compared to the case of parallel boundary conditions is the asymptotic behavior of the scattering coefficients, which affects the normalization of the eigenfunctions and the sectionally meromorphic matrix in the RHP. When the asymptotic polarization vectors are not orthogonal, two different methods are presented to convert the RHP into a set of linear algebraic-integral equations. When the asymptotic polarization vectors are orthogonal, however, only one of these methods is applicable. Finally, it is shown that, both in the case of orthogonal and non-orthogonal polarization vectors, no reflectionless potentials can exist, which implies that the problem does not admit pure soliton solutions.