Unique continuation property for a higher order nonlinear Schrödinger equation

We prove that, if a sufficiently smooth solution u to the initial value problem associated with the equation ∂tu+iα∂x2u+β∂x3u+iγ|u|2u+δ|u|2∂xu+εu2∂xu¯=0,x,t∈R, is supported in a half line at two different instants of time then u≡0. To prove this result we derive a new Carleman type estimate by extending the method introduced by Kenig et al. in [Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002) 191-208].