Construction of smooth chiral finite-time blow-up solutions to Calogero--Moser derivative nonlinear Schrödinger equation
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911141780783104 |
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| author | Kim, Kihyun Kim, Taegyu Kwon, Soonsik |
| author_facet | Kim, Kihyun Kim, Taegyu Kwon, Soonsik |
| contents | We consider the Calogero--Moser derivative nonlinear Schrödinger equation (CM-DNLS), which is an $L^{2}$-critical nonlinear Schrödinger equation with explicit solitons, self-duality, and pseudo-conformal symmetry. More importantly, this equation is known to be completely integrable in the Hardy space $L_{+}^{2}$ and the solutions in this class are referred to as \emph{chiral} solutions. A rigorous PDE analysis of this equation with complete integrability was recently initiated by Gérard and Lenzmann.
Our main result constructs smooth, chiral, and finite energy finite-time blow-up solutions with mass arbitrarily close to that of a soliton, answering the global regularity question for chiral solutions raised by Gérard and Lenzmann. The blow-up rate obtained for these solutions is different from the pseudo-conformal rate. Our proof also gives a construction of a codimension one set of smooth finite energy initial data (but without addressing chirality) leading to the same blow-up dynamics. Our blow-up construction in the Hardy space might also be contrasted with the global well-posedness of the derivative nonlinear Schrödinger equation (DNLS), which is another integrable $L^{2}$-critical Schrödinger equation.
The overall scheme of our proof is the forward construction of blow-up dynamics with modulation analysis. We begin with developing a linear theory for the near-soliton dynamics. We discover a nontrivial conjugation identity, which unveils a surprising connection from the linearized (CM-DNLS) to the 1D free Schrödinger equation, which is a crucial ingredient for overcoming the difficulties from the nonlocal nonlinearity. Another principal challenge in this work, the slow decay of the soliton, is overcome by introducing a trick of decomposing solutions depending on topologies, which we believe is of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09603 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Construction of smooth chiral finite-time blow-up solutions to Calogero--Moser derivative nonlinear Schrödinger equation Kim, Kihyun Kim, Taegyu Kwon, Soonsik Analysis of PDEs 35B44 (primary), 35Q55, 37K10, 37K40 We consider the Calogero--Moser derivative nonlinear Schrödinger equation (CM-DNLS), which is an $L^{2}$-critical nonlinear Schrödinger equation with explicit solitons, self-duality, and pseudo-conformal symmetry. More importantly, this equation is known to be completely integrable in the Hardy space $L_{+}^{2}$ and the solutions in this class are referred to as \emph{chiral} solutions. A rigorous PDE analysis of this equation with complete integrability was recently initiated by Gérard and Lenzmann. Our main result constructs smooth, chiral, and finite energy finite-time blow-up solutions with mass arbitrarily close to that of a soliton, answering the global regularity question for chiral solutions raised by Gérard and Lenzmann. The blow-up rate obtained for these solutions is different from the pseudo-conformal rate. Our proof also gives a construction of a codimension one set of smooth finite energy initial data (but without addressing chirality) leading to the same blow-up dynamics. Our blow-up construction in the Hardy space might also be contrasted with the global well-posedness of the derivative nonlinear Schrödinger equation (DNLS), which is another integrable $L^{2}$-critical Schrödinger equation. The overall scheme of our proof is the forward construction of blow-up dynamics with modulation analysis. We begin with developing a linear theory for the near-soliton dynamics. We discover a nontrivial conjugation identity, which unveils a surprising connection from the linearized (CM-DNLS) to the 1D free Schrödinger equation, which is a crucial ingredient for overcoming the difficulties from the nonlocal nonlinearity. Another principal challenge in this work, the slow decay of the soliton, is overcome by introducing a trick of decomposing solutions depending on topologies, which we believe is of independent interest. |
| title | Construction of smooth chiral finite-time blow-up solutions to Calogero--Moser derivative nonlinear Schrödinger equation |
| topic | Analysis of PDEs 35B44 (primary), 35Q55, 37K10, 37K40 |
| url | https://arxiv.org/abs/2404.09603 |