Asymptotic Behaviour for Schrodinger Equations with a Quadratic Nonlinearity in One-Space Dimension

Abstract

We consider the Cauchy problem for the Schrödinger equation with a quadratic nonlinearity in one space dimension iut + 1/2 uxx = t-α |ux|2, u(0, x) = u0(x), where α ∈ (0, 1). From the heuristic point of view, solutions to this problem should have a quasilinear character when α ∈ (1/2, 1). We show in this paper that the solutions do not have a quasilinear character for all α ∈ [1/2, 1) if the initial data u0 ∈ H3,0 ∩ H2,2 are small, then the solution has a slow time decay such as t-α/2. For α ∈ (0,1/2), if we assume that the initial data u0 are analytic and small, then the small time decay occurs.