Dynamics of solutions in the 1d bi-harmonic nonlinear Schrödinger equation
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918359858151424 |
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| author | Klein, Christian Petrenko, Iryna Roudenko, Svetlana Stoilov, Nikola |
| author_facet | Klein, Christian Petrenko, Iryna Roudenko, Svetlana Stoilov, Nikola |
| contents | We consider the one dimensional 4th order, or bi-harmonic, nonlinear Schrödinger (NLS) equation, namely, $i u_t - Δ^2 u - 2a Δu + |u|^α u = 0, ~ x,a \in \R$, $α>0$, and investigate the dynamics of its solutions for various powers of $α$, including the ground state solutions and their perturbations, leading to scattering or blow-up dichotomy when $a \leq 0$, or to a trichotomy when $a>0$. Ground state solutions are numerically constructed, and their stability is studied, finding that the ground state solutions may form two branches, stable and unstable, which dictates the long-term behavior of solutions. Perturbations of the ground states on the unstable branch either lead to dispersion or the jump to a stable ground state. In the critical and supercritical cases, blow-up in finite time is also investigated, and it is conjectured that the blow-up happens with a scale-invariant profile (when $a=0$) regardless of the value of $a$ of the lower dispersion. The blow-up rate is also explored. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_24961 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dynamics of solutions in the 1d bi-harmonic nonlinear Schrödinger equation Klein, Christian Petrenko, Iryna Roudenko, Svetlana Stoilov, Nikola Analysis of PDEs We consider the one dimensional 4th order, or bi-harmonic, nonlinear Schrödinger (NLS) equation, namely, $i u_t - Δ^2 u - 2a Δu + |u|^α u = 0, ~ x,a \in \R$, $α>0$, and investigate the dynamics of its solutions for various powers of $α$, including the ground state solutions and their perturbations, leading to scattering or blow-up dichotomy when $a \leq 0$, or to a trichotomy when $a>0$. Ground state solutions are numerically constructed, and their stability is studied, finding that the ground state solutions may form two branches, stable and unstable, which dictates the long-term behavior of solutions. Perturbations of the ground states on the unstable branch either lead to dispersion or the jump to a stable ground state. In the critical and supercritical cases, blow-up in finite time is also investigated, and it is conjectured that the blow-up happens with a scale-invariant profile (when $a=0$) regardless of the value of $a$ of the lower dispersion. The blow-up rate is also explored. |
| title | Dynamics of solutions in the 1d bi-harmonic nonlinear Schrödinger equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.24961 |