Bibliographic Details
| Title: |
On the compactness of the support of solitary waves of the complex saturated nonlinear Schr{\'o}dinger equation and related problems |
| Authors: |
Bégout, Pascal, Díaz, Jesús Ildefonso |
| Source: |
Physica D: Nonlinear Phenomena, 2025, 472, pp.134516 |
| Publication Year: |
2025 |
| Collection: |
Mathematics |
| Subject Terms: |
Mathematics - Analysis of PDEs |
| More Details: |
We study the vectorial stationary Schr\"odinger equation $-\Delta u+a\,U+b\,u=F,$ with a saturated nonlinearity $U=u/|u|$ and with some complex coefficients $(a,b)\in\mathbb{C}^2$. Besides the existence and uniqueness of solutions for the Dirichlet and Neumann problems, we prove the compactness of the support of the solution, under suitable conditions on $(a,b)$ and even when the source in the right hand side $F(x)$ is not vanishing for large values of $|x|.$ The proof of the compactness of the support uses a local energy method, given the impossibility of applying the maximum principle. We also consider the associate Schr\"{o}dinger-Poisson system when coupling with a simple magnetic field. Among other consequences, our results give a rigorous proof of the existence of ``solitons with compact support" claimed, without any proof, by several previous authors. |
| Document Type: |
Working Paper |
| DOI: |
10.1016/j.physd.2024.134516 |
| Access URL: |
http://arxiv.org/abs/2501.07181 |
| Accession Number: |
edsarx.2501.07181 |
| Database: |
arXiv |
