Abstract
In this paper, we study the fourth order nonlinear Schrödinger type equation (4NLS) which is a generalization of the Fukumoto-Moffatt [5] model that arising in the context of the motion of a vortex filament. Firstly, we mention the existence of standing wave solution and the conserved quantities. We next investigate the case that the equation is completely integrable and show that the standing wave obtained in [20] is orbitally stable in Sobolev spaces Hm with m ∈ N. Further, we show that the completely integrable equation is ill-posed in Hs with s ∈ (-1/2,1/2) by following Kenig-Ponce-Vega [13].
