Long–time Asymptotics for Solutions of the Nls Equation with Initial Data in a Weighted Sobolev Space

Here Γ is the gamma function and the function r is the so-called reflection coefficient for the potential q0(x) = q(x, t = 0), as described below. The error term O( log t t ) is uniform for all x ∈ R. The above asymptotic form was first obtained in [ZaMa], but without the error estimate. Based on the nonlinear steepest descent method introduced in [DZ1], the error estimate in (1.1) was derived in [DIZ] (see also [DZ2] for a pedagogic presentation). As noted above, some high orders of decay and smoothness are required for the initial data. In this paper we describe a new method that produces an error estimate of order O(t 1 2) for any 0 < κ < 1 4 , just under the assumption that the initial data q0 lies in the weighted Sobolev space H = {f ∈ L(R) : xf, f ′ ∈ L2(R)}. Such an estimate is needed, for example, in [DZ3][DZ5] where the authors obtain long-time asymptotics for solutions of the perturbed NLS equation, iqt+qxx−2|q|q−ǫ|q|q = 0, for l > 2 and ǫ > 0. As we will see (cf. Section 4 below), the estimate O(t 1 2) in fact depends only on the weighted L norm of the initial data, (∫