Quantized blow-up dynamics for Calogero--Moser derivative nonlinear Schrödinger equation
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| Format: | Preprint |
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2024
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| author | Jeong, Uihyeon Kim, Taegyu |
| author_facet | Jeong, Uihyeon Kim, Taegyu |
| contents | We consider the Calogero--Moser derivative nonlinear Schrödinger equation (CM-DNLS), an $L^2$-critical nonlinear Schrödinger type equation enjoying a number of numerous structures, such as nonlocal nonlinearity, self-duality, pseudo-conformal symmetry, and complete integrability.
In this paper, we construct smooth finite-time blow-up solutions to (CM-DNLS) that exhibit a sequence of discrete blow-up rates, so-called \emph{quantized blow-up rates}. Our strategy is a forward construction of the blow-up dynamics based on modulation analysis. Our main novelty is to utilize the \emph{nonlinear adapted derivative} suited to the \textit{Lax pair structure} and to rely on the \emph{hierarchy of conservation laws} inherent in this structure to control higher-order energies. This approach replaces a repulsivity-based energy method in the bootstrap argument, which significantly simplifies the analysis compared to earlier works. Our result highlights that the integrable structure remains a powerful tool, even in the presence of blow-up solutions.
In (CM-DNLS), one of the distinctive features is \emph{chirality}. However, our constructed solutions are not chiral, since we assume the radial (even) symmetry in the gauge transformed equation. This radial assumption simplifies the modulation analysis. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_12518 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantized blow-up dynamics for Calogero--Moser derivative nonlinear Schrödinger equation Jeong, Uihyeon Kim, Taegyu Analysis of PDEs 35B44 (primary), 35Q55, 37K10 We consider the Calogero--Moser derivative nonlinear Schrödinger equation (CM-DNLS), an $L^2$-critical nonlinear Schrödinger type equation enjoying a number of numerous structures, such as nonlocal nonlinearity, self-duality, pseudo-conformal symmetry, and complete integrability. In this paper, we construct smooth finite-time blow-up solutions to (CM-DNLS) that exhibit a sequence of discrete blow-up rates, so-called \emph{quantized blow-up rates}. Our strategy is a forward construction of the blow-up dynamics based on modulation analysis. Our main novelty is to utilize the \emph{nonlinear adapted derivative} suited to the \textit{Lax pair structure} and to rely on the \emph{hierarchy of conservation laws} inherent in this structure to control higher-order energies. This approach replaces a repulsivity-based energy method in the bootstrap argument, which significantly simplifies the analysis compared to earlier works. Our result highlights that the integrable structure remains a powerful tool, even in the presence of blow-up solutions. In (CM-DNLS), one of the distinctive features is \emph{chirality}. However, our constructed solutions are not chiral, since we assume the radial (even) symmetry in the gauge transformed equation. This radial assumption simplifies the modulation analysis. |
| title | Quantized blow-up dynamics for Calogero--Moser derivative nonlinear Schrödinger equation |
| topic | Analysis of PDEs 35B44 (primary), 35Q55, 37K10 |
| url | https://arxiv.org/abs/2412.12518 |