Kodai Mathematical Journal

Stability of line standing waves near the bifurcation point for nonlinear Schrödinger equations

Yohei Yamazaki

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Abstract

In this paper we consider the transverse instability for a nonlinear Schrödinger equation with power nonlinearity on R × TL, where 2πL is the period of the torus TL. There exists a critical period 2πLω,p such that the line standing wave is stable for L < Lω,p and the line standing wave is unstable for L > Lω,p. Here we farther study the bifurcation from the boundary L = Lω,p between the stability and the instability for line standing waves of the nonlinear Schrödinger equation. We show the stability for the branch bifurcating from the line standing waves by applying the argument in Kirr, Kevrekidis and Pelinovsky [16] and the method in Grillakis, Shatah and Strauss [12]. However, at the bifurcation point, the linearized operator around the bifurcation point is degenerate. To prove the stability for the bifurcation point, we apply the argument in Maeda [18].

Article information

Source
Kodai Math. J., Volume 38, Number 1 (2015), 65-96.

Dates
First available in Project Euclid: 18 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.kmj/1426684443

Digital Object Identifier
doi:10.2996/kmj/1426684443

Mathematical Reviews number (MathSciNet)
MR3323514

Zentralblatt MATH identifier
1323.35167

Citation

Yamazaki, Yohei. Stability of line standing waves near the bifurcation point for nonlinear Schrödinger equations. Kodai Math. J. 38 (2015), no. 1, 65--96. doi:10.2996/kmj/1426684443. https://projecteuclid.org/euclid.kmj/1426684443


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