Asymptotic Behaviour for Schrodinger Equations with a Quadratic Nonlinearity in One-Space Dimension
Date
2001-07-25
Authors
Hayashi, Nakao
Naumkin, Pavel I.
Journal Title
Journal ISSN
Volume Title
Publisher
Southwest Texas State University, Department of Mathematics
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.
Description
Keywords
Schrodinger equation, Large time behaviour, Quadratic nonlinearity
Citation
Hayashi, N., & Naumkin, P. I. (2001). Asymptotic behaviour for Schrodinger equations with a quadratic nonlinearity in one-space dimension. <i>Electronic Journal of Differential Equations, 2001</i>(54), pp. 1-18.
Rights
Attribution 4.0 International