Solvability of the Zakharov-Shabat systems with meromorphic potentials by quadrature
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913938738774016 |
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| author | Yagasaki, Kazuyuki |
| author_facet | Yagasaki, Kazuyuki |
| contents | We study the solvability of the general two-dimensional Zakharov-Shabat (ZS) systems with meromorphic potentials by quadrature. These systems appear in application of the inverse scattering transform (IST) to an important class of nonlinear partial differential equations (PDEs) called integrable systems. Their solvability by quadrature is a key to obtain analytical expressions for solutions to the initial value problems of the integrable PDEs by using the IST. We prove that the ZS systems are always integrable in the sense of differential Galois theory, i.e., solvable by quadrature, if and only if the meromporphic potentials are reflectionless, under the condition that the potentials are absolutely integrable on $\mathbb{R}\setminus(-R_0,R_0)$ for some $R_0>0$. Similar statements were previously proved to be true by the author for a limited class of potentials and the linear Schrödinger equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_07246 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solvability of the Zakharov-Shabat systems with meromorphic potentials by quadrature Yagasaki, Kazuyuki Analysis of PDEs Dynamical Systems 35Q51, 37K15, 34M03, 34M15, 34M35, 34M40, 35P25, 37K40 We study the solvability of the general two-dimensional Zakharov-Shabat (ZS) systems with meromorphic potentials by quadrature. These systems appear in application of the inverse scattering transform (IST) to an important class of nonlinear partial differential equations (PDEs) called integrable systems. Their solvability by quadrature is a key to obtain analytical expressions for solutions to the initial value problems of the integrable PDEs by using the IST. We prove that the ZS systems are always integrable in the sense of differential Galois theory, i.e., solvable by quadrature, if and only if the meromporphic potentials are reflectionless, under the condition that the potentials are absolutely integrable on $\mathbb{R}\setminus(-R_0,R_0)$ for some $R_0>0$. Similar statements were previously proved to be true by the author for a limited class of potentials and the linear Schrödinger equations. |
| title | Solvability of the Zakharov-Shabat systems with meromorphic potentials by quadrature |
| topic | Analysis of PDEs Dynamical Systems 35Q51, 37K15, 34M03, 34M15, 34M35, 34M40, 35P25, 37K40 |
| url | https://arxiv.org/abs/2506.07246 |