# American Institute of Mathematical Sciences

May  2015, 14(3): 843-859. doi: 10.3934/cpaa.2015.14.843

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

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

Received  April 2014 Revised  December 2014 Published  March 2015

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).
Citation: Jun-ichi Segata. Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves. Communications on Pure & Applied Analysis, 2015, 14 (3) : 843-859. doi: 10.3934/cpaa.2015.14.843
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##### References:
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