Solutions for fourth order anisotropic nonlinear Schrödinger equations in $\R^2$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909227163844608 |
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| author | Georgiev, Vladimir Gou, Tianxiang |
| author_facet | Georgiev, Vladimir Gou, Tianxiang |
| contents | In this paper, we consider solutions to the following fourth order anisotropic nonlinear Schrödinger equation in $\R \times \R^2$, $$ \left\{ \begin{aligned} &\textnormal{i}\partial_tψ+\partial_{xx} ψ-\partial_{yyyy} ψ+|ψ|^{p-2} ψ=0, \\ &ψ(0)=ψ_0 \in H^{1,2}(\R^2), \end{aligned} \right. $$ where $p>2$. First we prove the local/global well-posedness and blowup of solutions to the Cauchy problem for the anisotropic nonlinear Schrödinger equation. Then we establish the existence, axial symmetry, exponential decay and orbital stability/instability of standing waves to the anisotropic nonlinear Schrödinger equation. The pictures are considerably different from the ones for the isotropic nonlinear Schrödinger equations. The results are easily extendable to the higher dimensional case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12988 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Solutions for fourth order anisotropic nonlinear Schrödinger equations in $\R^2$ Georgiev, Vladimir Gou, Tianxiang Analysis of PDEs 35A15, 35Q55 In this paper, we consider solutions to the following fourth order anisotropic nonlinear Schrödinger equation in $\R \times \R^2$, $$ \left\{ \begin{aligned} &\textnormal{i}\partial_tψ+\partial_{xx} ψ-\partial_{yyyy} ψ+|ψ|^{p-2} ψ=0, \\ &ψ(0)=ψ_0 \in H^{1,2}(\R^2), \end{aligned} \right. $$ where $p>2$. First we prove the local/global well-posedness and blowup of solutions to the Cauchy problem for the anisotropic nonlinear Schrödinger equation. Then we establish the existence, axial symmetry, exponential decay and orbital stability/instability of standing waves to the anisotropic nonlinear Schrödinger equation. The pictures are considerably different from the ones for the isotropic nonlinear Schrödinger equations. The results are easily extendable to the higher dimensional case. |
| title | Solutions for fourth order anisotropic nonlinear Schrödinger equations in $\R^2$ |
| topic | Analysis of PDEs 35A15, 35Q55 |
| url | https://arxiv.org/abs/2406.12988 |