Mechanisms of unstable blowup in a quadratic nonlinear Schrödinger equation
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914968834670592 |
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| author | Jaquette, Jonathan |
| author_facet | Jaquette, Jonathan |
| contents | In the work Cho et al. [Jpn. J. Ind. Appl. Math. 33 (2016): 145-166] the authors conjecture that the quadratic nonlinear Schrödinger equation (NLS) $i u_t = u_{xx} + u^2 $ for $ x \in \mathbb{T}$ is globally well-posed for real initial data. We identify initial data whose numerical solution blows up in contradiction of this conjecture. The solution exhibits self-similar blowup and potentially nontrivial self-similar dynamics, however the proper scaling ansatz remains elusive.
Furthermore, the set of real initial data which blows up under the NLS dynamics appears to occur on a codimension-1 manifold, and we conjecture that it is precisely the stable manifold of the zero equilibrium for the nonlinear heat equation $u_t = u_{xx} + u^2 $. We apply the parameterization method to study the internal dynamics of this manifold, offering a heuristic argument in support of our conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00762 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mechanisms of unstable blowup in a quadratic nonlinear Schrödinger equation Jaquette, Jonathan Analysis of PDEs Dynamical Systems In the work Cho et al. [Jpn. J. Ind. Appl. Math. 33 (2016): 145-166] the authors conjecture that the quadratic nonlinear Schrödinger equation (NLS) $i u_t = u_{xx} + u^2 $ for $ x \in \mathbb{T}$ is globally well-posed for real initial data. We identify initial data whose numerical solution blows up in contradiction of this conjecture. The solution exhibits self-similar blowup and potentially nontrivial self-similar dynamics, however the proper scaling ansatz remains elusive. Furthermore, the set of real initial data which blows up under the NLS dynamics appears to occur on a codimension-1 manifold, and we conjecture that it is precisely the stable manifold of the zero equilibrium for the nonlinear heat equation $u_t = u_{xx} + u^2 $. We apply the parameterization method to study the internal dynamics of this manifold, offering a heuristic argument in support of our conjecture. |
| title | Mechanisms of unstable blowup in a quadratic nonlinear Schrödinger equation |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2406.00762 |