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 ∂t u +iα∂2 x u + β∂3 x u + iγ |u|2u +δ|u|2∂xu + u2∂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].  2004 Elsevier Inc. All rights reserved