Stability of Hasimoto solitons in energy space for a fourth order nonlinear Schrödinger type equation

Zhong Wang

doi:10.3934/dcds.2017174

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2017, Volume 37, Issue 7: 4091-4108. Doi: 10.3934/dcds.2017174

This issue Previous Article Dacorogna-Moser theorem on the Jacobian determinant equation with control of support Next Article Gradient estimates for the strong $p(x)$-Laplace equation

Stability of Hasimoto solitons in energy space for a fourth order nonlinear Schrödinger type equation

Zhong Wang^,

School of Mathematics and Big Data, Foshan University, Foshan, Guangdong 528000, China

* Corresponding author: Zhong Wang.
* Corresponding author: Zhong Wang.

Received: November 30, 2015

Revised: March 31, 2016

Published: August 2017

This work is supported by the China National Natural Science Foundation under grant number 11571381

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In this article we investigate the nonlinear stability of Hasimoto solitons, in energy space, for a fourth order Schrödinger equation (4NLS) which arises in the context of the vortex filament. The proof relies on a suitable Lyapunov functional, at the $H^2$ level, which allows us to describe the dynamics of small perturbations. This stability result is also extended to Sobolev spaces $H^m$ for all $m∈\mathbb{Z}_+$ by employing the infinite conservation laws of 4NLS.

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Mathematics Subject Classification: Primary: 35Q35, 76W05; Secondary: 35B65.

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