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

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 H²_per (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). Fourth order nonlinear Schrödinger type equation, stability of standing waves.