New Lower Bounds on the Radius of Spatial Analyticity for the Third-Order Nonlinear Schrödinger Equation

In this paper, we consider the initial value problem associated with the cubic nonlinear Schrödinger equation with third-order dispersion ∂tu+iα∂x2u+β∂x3u+iγu2u=0,x∈℠,t∈℠,ux,0=u0x, where α,β and γ are real constants such that β,γ≠0, u is a complex valued function and the initial data u0 is analytic on ℠and has a uniform radius of analyticity σ0 in the spatial variable. We show the uniform radius of spatial analyticity σT of solutions at time T cannot decay faster than cT−2 for large T. This improves a recent result by Figueira and Panthee, where they established a decay rate of order cT−4+ε for ε>0. We used an approximate conservation law in the modified Gevrey spaces, a contraction mapping argument, space-time estimates and one-dimensional Sobolev embedding to obtain our improved result.