Classification and Nondegeneracy of Cubic Nonlinear Schrödinger System in $\mathbb{R}$
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| Formato: | Preprint |
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2024
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| _version_ | 1866914474571595776 |
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| author | Guo, Yujin Luo, Yong Wei, Juncheng |
| author_facet | Guo, Yujin Luo, Yong Wei, Juncheng |
| contents | We study the following one-dimensional cubic nonlinear Schrödinger system: \[ u_i''+2\Big(\sum_{k=1}^Nu_k^2\Big)u_i=-μ_iu_i \ \,\ \mbox{in}\, \ \mathbb{R} , \ \ i=1, 2, \cdots, N, \] where $μ_1\leqμ_2\leq\cdots\leqμ_N<0$ and $N\ge 2$. In this paper, we mainly focus on the case $N=3$ and prove the following results: (i). The solutions of the system can be completely classified; (ii). Depending on the explicit values of $μ_1\leqμ_2\leqμ_3<0$, there exist two different classes of normalized solutions $u=(u_1, u_2, u_3)$ satisfying $\int _{R}u_i^2dx=1$ for all $i=1, 2, 3$, which are completely different from the case $N=2$; (iii). The linearized operator at any nontrivial solution of the system is non-degenerate. The conjectures on the explicit classification and nondegeneracy of solutions for the system are also given for the case $N>3$. These address the questions of [R. Frank, D. Gontier and M. Lewin, CMP, 2021], where the complete classification and uniqueness results for the system were already proved for the case $N=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10748 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Classification and Nondegeneracy of Cubic Nonlinear Schrödinger System in $\mathbb{R}$ Guo, Yujin Luo, Yong Wei, Juncheng Analysis of PDEs We study the following one-dimensional cubic nonlinear Schrödinger system: \[ u_i''+2\Big(\sum_{k=1}^Nu_k^2\Big)u_i=-μ_iu_i \ \,\ \mbox{in}\, \ \mathbb{R} , \ \ i=1, 2, \cdots, N, \] where $μ_1\leqμ_2\leq\cdots\leqμ_N<0$ and $N\ge 2$. In this paper, we mainly focus on the case $N=3$ and prove the following results: (i). The solutions of the system can be completely classified; (ii). Depending on the explicit values of $μ_1\leqμ_2\leqμ_3<0$, there exist two different classes of normalized solutions $u=(u_1, u_2, u_3)$ satisfying $\int _{R}u_i^2dx=1$ for all $i=1, 2, 3$, which are completely different from the case $N=2$; (iii). The linearized operator at any nontrivial solution of the system is non-degenerate. The conjectures on the explicit classification and nondegeneracy of solutions for the system are also given for the case $N>3$. These address the questions of [R. Frank, D. Gontier and M. Lewin, CMP, 2021], where the complete classification and uniqueness results for the system were already proved for the case $N=2$. |
| title | Classification and Nondegeneracy of Cubic Nonlinear Schrödinger System in $\mathbb{R}$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.10748 |