Decay of solutions to one-dimensional inhomogeneous nonlinear Schrödinger equations
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915597763215360 |
|---|---|
| author | Cui, Zhi-Yuan Li, Yuan Zhao, Dun |
| author_facet | Cui, Zhi-Yuan Li, Yuan Zhao, Dun |
| contents | We investigate the decay estimates of global solutions for a class of one-dimensional inhomogeneous nonlinear Schrödinger equations. While most existing results focus on spatial dimensions $d\geq2$, the decay properties in one dimension remain less explored due to the absence of effective Morawetz inequalities. For equations without external potential, by establishing a localized Virial-Morawetz identity, we derive decay estimates in the context of the $L^r$-norm for global solutions within a compact domain as a time subsequence approaches infinity. This decay result can be applied to obtain a criterion for energy scattering. Additionally, by establishing another type of Virial-Morawetz identity under more strict conditions, we demonstrate the decay result for odd solutions for any time sequence that approaches infinity. Utilizing some results about bound states proved by Barry Simon, we also show that similar decay results hold for the global odd solutions of equations with suitable external potentials that contain inverse power type and Yukawa-type potentials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_08272 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Decay of solutions to one-dimensional inhomogeneous nonlinear Schrödinger equations Cui, Zhi-Yuan Li, Yuan Zhao, Dun Analysis of PDEs 35Q55, 35B40 We investigate the decay estimates of global solutions for a class of one-dimensional inhomogeneous nonlinear Schrödinger equations. While most existing results focus on spatial dimensions $d\geq2$, the decay properties in one dimension remain less explored due to the absence of effective Morawetz inequalities. For equations without external potential, by establishing a localized Virial-Morawetz identity, we derive decay estimates in the context of the $L^r$-norm for global solutions within a compact domain as a time subsequence approaches infinity. This decay result can be applied to obtain a criterion for energy scattering. Additionally, by establishing another type of Virial-Morawetz identity under more strict conditions, we demonstrate the decay result for odd solutions for any time sequence that approaches infinity. Utilizing some results about bound states proved by Barry Simon, we also show that similar decay results hold for the global odd solutions of equations with suitable external potentials that contain inverse power type and Yukawa-type potentials. |
| title | Decay of solutions to one-dimensional inhomogeneous nonlinear Schrödinger equations |
| topic | Analysis of PDEs 35Q55, 35B40 |
| url | https://arxiv.org/abs/2412.08272 |