Unique continuation properties of the higher order nonlinear Schrödinger equations in one dimension

In this paper, we study the unique continuation properties of the higher order nonlinear Schrödinger equations in one dimension and show exponential decay weighted estimates, as well as a Lp-type Carleman estimate based on the Littlewood-Paley theory. As a consequence we obtain that if u is a solution of the Schrödinger equation such that there exists X0≠±∞ with suppu(⋅,0), suppu(⋅,1)⊆(−∞,X0) (or (X0,+∞)) then u≡0 in R×[0,1].