Abstract
In this paper, the existence and orbital stability of the periodic standing wave solutions for the nonlinear fractional Schrödinger (fNLS) equation with cubic nonlinearity is studied. The existence is determined by using a minimizing constrained problem in the complex setting and it is showed that the corresponding real solution is always positive. The orbital stability is proved by combining some tools regarding the oscillation theorem for fractional Hill operators and the Vakhitov-Kolokolov condition, well known for Schrödinger equations. We then perform a numerical approach to generate the periodic standing wave solutions of the fNLS equation by using the Petviashvili's iteration method. We also investigate the Vakhitov-Kolokolov condition numerically which cannot be obtained analytically for some values of the order of the fractional derivative.
| Original language | English |
|---|---|
| Pages (from-to) | 263-291 |
| Number of pages | 29 |
| Journal | Journal of Differential Equations |
| Volume | 341 |
| DOIs | |
| Publication status | Published - 25 Dec 2022 |
Bibliographical note
Publisher Copyright:© 2022 Elsevier Inc.
Funding
G. de Loreno and G.E.B. Moraes are supported by the regular doctorate scholarship from CAPES/Brazil. F. Natali is partially supported by Fundação Araucária /Brazil (grant 002/2017 ), CNPq /Brazil (grant 303907/2021-5 ) and CAPES MathAmSud (grant 88881.520205/2020-01 ).
| Funders | Funder number |
|---|---|
| CAPES MathAmSud | 88881.520205/2020-01 |
| Coordenação de Aperfeiçoamento de Pessoal de Nível Superior | |
| Conselho Nacional de Desenvolvimento Científico e Tecnológico | 303907/2021-5 |
| Fundação Araucária | 002/2017 |
Keywords
- Existence and uniqueness of minimizers
- Fractional Schrödinger equation
- Orbital stability
- Small-amplitude periodic waves