Initial value problem for the fourth ordernonlinear Schrödinger type equationon torus and orbital stability ofstanding waves

Jun-ichi Segata

doi:10.3934/cpaa.2015.14.843

Article Contents

2015, Volume 14, Issue 3: 843-859. Doi: 10.3934/cpaa.2015.14.843

This issue Previous Article Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations Next Article Admissibility, a general type of Lipschitz shadowing and structural stability

Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves

Jun-ichi Segata^1,

1.
Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578

Received: April 2014

Revised: December 2014

Published: May 2015

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Abstract

We consider the fourth order nonlinear Schrödinger type equation (4NLS) which arises in context of the motion of vortex filament. The purposes of this paper are twofold. Firstly, we consider the initial value problem for (4NLS) under the periodic boundary condition. By refining the modified energy method used in our previous paper [23], we prove the unique existence of the global solution for (4NLS) in the energy space $H_{p e r}^2(0,2L)$ with $L>0$. Secondly, we study the stability property of periodic standing waves for (4NLS). Using the spectrum properties of the Schrödinger operators associated to the periodic standing wave developed by Angulo [1], we prove that standing wave of dnoidal type is orbitally stable under the time evolution by (4NLS).

Keywords:

Fourth order nonlinear Schrödinger type equation,
stability of standing waves.

Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B35, 35B40.

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References

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