Soliton resolution for equivariant self-dual Chern-Simons-Schrödinger equation in weighted Sobolev class
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arXiv
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| Format: | Preprint |
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2022
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| author | Kim, Kihyun Kwon, Soonsik Oh, Sung-Jin |
| author_facet | Kim, Kihyun Kwon, Soonsik Oh, Sung-Jin |
| contents | We consider the self-dual Chern-Simons-Schrödinger equation (CSS) under equivariant symmetry, which is a $L^{2}$-critical equation. It is known that (CSS) admits solitons and finite-time blow-up solutions. In this paper, we show soliton resolution for any solutions with equivariant data in the weighted Sobolev space $H^{1,1}$: every maximal solution decomposes into at most one modulated soliton and a radiation. A striking fact is that the nonscattering part must be a single modulated soliton. To our knowledge, this is the first result on soliton resolution in a class of nonlinear Schrödinger equations which are not known to be completely integrable. The key ingredient is the defocusing nature of the equation in the exterior of a soliton profile. This is a consequence of two distinctive features of (CSS): self-duality and non-local nonlinearity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_07314 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Soliton resolution for equivariant self-dual Chern-Simons-Schrödinger equation in weighted Sobolev class Kim, Kihyun Kwon, Soonsik Oh, Sung-Jin Analysis of PDEs 35B40, 35Q55, 37K40 We consider the self-dual Chern-Simons-Schrödinger equation (CSS) under equivariant symmetry, which is a $L^{2}$-critical equation. It is known that (CSS) admits solitons and finite-time blow-up solutions. In this paper, we show soliton resolution for any solutions with equivariant data in the weighted Sobolev space $H^{1,1}$: every maximal solution decomposes into at most one modulated soliton and a radiation. A striking fact is that the nonscattering part must be a single modulated soliton. To our knowledge, this is the first result on soliton resolution in a class of nonlinear Schrödinger equations which are not known to be completely integrable. The key ingredient is the defocusing nature of the equation in the exterior of a soliton profile. This is a consequence of two distinctive features of (CSS): self-duality and non-local nonlinearity. |
| title | Soliton resolution for equivariant self-dual Chern-Simons-Schrödinger equation in weighted Sobolev class |
| topic | Analysis of PDEs 35B40, 35Q55, 37K40 |
| url | https://arxiv.org/abs/2202.07314 |