Advances in Differential Equations

Transverse instability for nonlinear Schrödinger equation with a linear potential

Yohei Yamazaki

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Abstract

In this paper, we consider the transverse instability for a nonlinear Schrödinger equation with a linear potential on ${\mathbb {R} \times \mathbb {T}_L}$, where $2\pi L$ is the period of the torus $\mathbb{T}_L$. Rose and Weinstein [18] showed the existence of a stable standing wave for a nonlinear Schrödinger equation with a linear potential. We regard the standing wave of nonlinear Schrödinger equation on ${\mathbb R}$ as a line standing wave of nonlinear Schrödinger equation on ${\mathbb R} \times {\mathbb T}_L$. We show the stability of line standing waves for all $L>0$ by using the argument of the previous paper [26].

Article information

Source
Adv. Differential Equations Volume 21, Number 5/6 (2016), 429-462.

Dates
First available in Project Euclid: 9 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.ade/1457536497

Mathematical Reviews number (MathSciNet)
MR3473581

Subjects
Primary: 35B32: Bifurcation [See also 37Gxx, 37K50] 35B35: Stability 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

Citation

Yamazaki, Yohei. Transverse instability for nonlinear Schrödinger equation with a linear potential. Adv. Differential Equations 21 (2016), no. 5/6, 429--462. https://projecteuclid.org/euclid.ade/1457536497.


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