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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2412.00413 |
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| _version_ | 1866915042292662272 |
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| author | Masaki, Satoshi |
| author_facet | Masaki, Satoshi |
| contents | In this article, we investigate the large-time behavior of small solutions to a system of one-dimensional cubic nonlinear Schrödinger equations with two components. In previous studies, a structural condition on the nonlinearity has been employed to guarantee the existence of a coercive, mass-type conserved quantity. We identify a new class of systems that do not satisfy such a condition and thus lack a coercive conserved quantity. Nonetheless, we establish the global existence and describe the large-time behavior for small solutions in this class. In this setting, the asymptotic profile is described in terms of solutions to the corresponding system of ordinary differential equations (ODEs). A key element of our analysis is the use of a quartic conserved quantity associated with the ODE system. Moreover, for a specific example within this class, we solve the ODE system explicitly, showing that the asymptotic behavior is expressed using Jacobi elliptic functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00413 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global existence and large-time behavior of solutions to cubic nonlinear Schrödinger systems without coercive conserved quantity Masaki, Satoshi Analysis of PDEs In this article, we investigate the large-time behavior of small solutions to a system of one-dimensional cubic nonlinear Schrödinger equations with two components. In previous studies, a structural condition on the nonlinearity has been employed to guarantee the existence of a coercive, mass-type conserved quantity. We identify a new class of systems that do not satisfy such a condition and thus lack a coercive conserved quantity. Nonetheless, we establish the global existence and describe the large-time behavior for small solutions in this class. In this setting, the asymptotic profile is described in terms of solutions to the corresponding system of ordinary differential equations (ODEs). A key element of our analysis is the use of a quartic conserved quantity associated with the ODE system. Moreover, for a specific example within this class, we solve the ODE system explicitly, showing that the asymptotic behavior is expressed using Jacobi elliptic functions. |
| title | Global existence and large-time behavior of solutions to cubic nonlinear Schrödinger systems without coercive conserved quantity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.00413 |