Short Time Behavior of Solutions to Nonlinear Schrödinger Equations in One and Two Space Dimensions

Abstract

We examine the behavior for small times of solutions to certain nonlinear Schrödinger equations u t = iΔ u + q(u), with initial data u(0) = u 0, with emphasis on cases where u 0 is compactly supported and piecewise smooth, with jump discontinuity, on ℝ n , with n = 1 or 2. A main point is to provide a first order correction to the solution to the corresponding linear equation, and to estimate the error of this approximation. The error estimates are more than adequate to display nonlinear variants of the Gibbs phenomenon (a result first established by DiFranco, 2004 for the defocusing cubic NLS, when n = 1) and also, when n = 2, a nonlinear variant of the Pinsky phenomenon.