Solvability of the Korteweg-de Vries equation under meromorphic initial conditions by quadrature
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913938745065472 |
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| author | Yagasaki, Kazuyuki |
| author_facet | Yagasaki, Kazuyuki |
| contents | We study the solvability of the Korteweg-de Vries equation under meromorphic initial conditions by quadrature when the inverse scattering transform (IST) is applied. It is a key to solve the Schrödinger equation appearing in the Lax pair in application of the IST. We show that the Schrödinger equation is always integrable in the sense of differential Galois theory, i.e., solvable by quadrature, if and only if the meromporphic potential is reflectionless, under the condition that the potential is absolutely integrable on $\mathbb{R}\setminus(-R_0,R_0)$ for some $R_0>0$.This statement was previously proved to be true by the author for a limited class of potentials. We also show that the Schrödinger equation is not integrable in this sense for rational potentials that decay at infinity but do not satisfy the weak condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_08046 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solvability of the Korteweg-de Vries equation under meromorphic initial conditions by quadrature Yagasaki, Kazuyuki Analysis of PDEs Dynamical Systems Exactly Solvable and Integrable Systems 35Q53, 37K15, 34M03, 34M15, 34M35, 34M40, 35P25 We study the solvability of the Korteweg-de Vries equation under meromorphic initial conditions by quadrature when the inverse scattering transform (IST) is applied. It is a key to solve the Schrödinger equation appearing in the Lax pair in application of the IST. We show that the Schrödinger equation is always integrable in the sense of differential Galois theory, i.e., solvable by quadrature, if and only if the meromporphic potential is reflectionless, under the condition that the potential is absolutely integrable on $\mathbb{R}\setminus(-R_0,R_0)$ for some $R_0>0$.This statement was previously proved to be true by the author for a limited class of potentials. We also show that the Schrödinger equation is not integrable in this sense for rational potentials that decay at infinity but do not satisfy the weak condition. |
| title | Solvability of the Korteweg-de Vries equation under meromorphic initial conditions by quadrature |
| topic | Analysis of PDEs Dynamical Systems Exactly Solvable and Integrable Systems 35Q53, 37K15, 34M03, 34M15, 34M35, 34M40, 35P25 |
| url | https://arxiv.org/abs/2506.08046 |