Nonlinear Instability of Half-Solitons on Star Graphs

We consider a half-soliton stationary state of the nonlinear Schrodinger equation with the power nonlinearity on a star graph consisting of N edges and a single vertex. For the subcritical power nonlinearity, the half-soliton state is a degenerate critical point of the action functional under the ma...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2017-05
Main Authors: Kairzhan, Adilbek, Pelinovsky, Dmitry E
Format: Article
Language:English
Subjects:
Critical point
Graph theory
Nonlinear equations
Nonlinearity
Saddle points
Schrodinger equation
Solitary waves
Stability
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We consider a half-soliton stationary state of the nonlinear Schrodinger equation with the power nonlinearity on a star graph consisting of N edges and a single vertex. For the subcritical power nonlinearity, the half-soliton state is a degenerate critical point of the action functional under the mass constraint such that the second variation is nonnegative. By using normal forms, we prove that the degenerate critical point is a nonlinear saddle point, for which the small perturbations to the half-soliton state grow slowly in time resulting in the nonlinear instability of the half-soliton state. The result holds for any \(N \geq 3\) and arbitrary subcritical power nonlinearity. It gives a precise dynamical characterization of the previous result of Adami {\em et al.}, where the half-soliton state was shown to be a saddle point of the action functional under the mass constraint for \(N = 3\) and for cubic nonlinearity.
ISSN:2331-8422