Taiwanese Journal of Mathematics

Quasi-periodic Solutions for Nonlinear Schrödinger Equations with Legendre Potential

Guanghua Shi and Dongfeng Yan

Full-text: Open access

Abstract

In this paper, the nonlinear Schrödinger equations with Legendre potential $\mathbf{i} u_{t} - u_{xx} + V_L(x)u + mu + \sec x \cdot |u|^2 u = 0$ subject to certain boundary conditions is considered, where $V_L(x) = -\frac{1}{2} - \frac{1}{4} \tan^2 x$, $x \in (-\pi/2,\pi/2)$. It is proved that for each given positive constant $m \gt 0$, the above equation admits lots of quasi-periodic solutions with two frequencies. The proof is based on a partial Birkhoff normal form technique and an infinite-dimensional Kolmogorov-Arnold-Moser theory.

Article information

Source
Taiwanese J. Math., Volume 24, Number 3 (2020), 663-679.

Dates
Received: 10 August 2018
Revised: 3 May 2019
Accepted: 28 July 2019
First available in Project Euclid: 19 May 2020

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1589875223

Digital Object Identifier
doi:10.11650/tjm/190707

Subjects
Primary: 37K55: Perturbations, KAM for infinite-dimensional systems 37C55: Periodic and quasiperiodic flows and diffeomorphisms

Keywords
Kolmogorov-Arnold-Moser theory quasi-periodic solutions singular differential operator

Citation

Shi, Guanghua; Yan, Dongfeng. Quasi-periodic Solutions for Nonlinear Schrödinger Equations with Legendre Potential. Taiwanese J. Math. 24 (2020), no. 3, 663--679. doi:10.11650/tjm/190707. https://projecteuclid.org/euclid.twjm/1589875223


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