Soliton resolution for Calogero--Moser derivative nonlinear Schrödinger equation
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arXiv
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| Format: | Preprint |
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2024
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| author | Kim, Taegyu Kwon, Soonsik |
| author_facet | Kim, Taegyu Kwon, Soonsik |
| contents | We consider soliton resolution for the Calogero--Moser derivative nonlinear Schrödinger equation (CM-DNLS). A rigorous PDE analysis of (CM-DNLS) was recently initiated by Gérard and Lenzmann, who demonstrated its Lax pair structure. Additionally, (CM-DNLS) exhibits several symmetries, such as mass-criticality with pseudo-conformal symmetry and a self-dual Hamiltonian. Despite its integrability, finite-time blow-up solutions have been constructed.
The purpose of this paper is to establish soliton resolution for both finite-time blow-up solutions and global solutions in a fully general setting, \emph{without imposing radial symmetry or size constraints}. To our knowledge, this is the first non-integrable proof of full soliton resolution for Schrödinger-type equations. A key aspect of our proof is the control of the energy of the outer radiation after extracting a soliton, referred to as the \emph{energy bubbling} estimate. This benefits from two levels of convervation laws, mass and energy, and self-duality. This approach allows us to directly prove continuous-in-time soliton resolution, bypassing time-sequential soliton resolution. Importantly, our proof does not rely on the integrability of the equation, potentially offering insights applicable to other non-integrable models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12843 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Soliton resolution for Calogero--Moser derivative nonlinear Schrödinger equation Kim, Taegyu Kwon, Soonsik Analysis of PDEs 35B40 (primary), 35Q55, 37K10, 37K40 We consider soliton resolution for the Calogero--Moser derivative nonlinear Schrödinger equation (CM-DNLS). A rigorous PDE analysis of (CM-DNLS) was recently initiated by Gérard and Lenzmann, who demonstrated its Lax pair structure. Additionally, (CM-DNLS) exhibits several symmetries, such as mass-criticality with pseudo-conformal symmetry and a self-dual Hamiltonian. Despite its integrability, finite-time blow-up solutions have been constructed. The purpose of this paper is to establish soliton resolution for both finite-time blow-up solutions and global solutions in a fully general setting, \emph{without imposing radial symmetry or size constraints}. To our knowledge, this is the first non-integrable proof of full soliton resolution for Schrödinger-type equations. A key aspect of our proof is the control of the energy of the outer radiation after extracting a soliton, referred to as the \emph{energy bubbling} estimate. This benefits from two levels of convervation laws, mass and energy, and self-duality. This approach allows us to directly prove continuous-in-time soliton resolution, bypassing time-sequential soliton resolution. Importantly, our proof does not rely on the integrability of the equation, potentially offering insights applicable to other non-integrable models. |
| title | Soliton resolution for Calogero--Moser derivative nonlinear Schrödinger equation |
| topic | Analysis of PDEs 35B40 (primary), 35Q55, 37K10, 37K40 |
| url | https://arxiv.org/abs/2408.12843 |