Electronic Journal of Differential Equations,
Vol. 2001(2001), No. 54, pp. 1-18.
Title: Asymptotic behaviour for Schrodinger equations with a quadratic
nonlinearity in one-space dimension
Author: Nakao Hayashi (Osaka Univ., Osaka, Japan)
Pavel I. Naumkin (Univ. Michoacana, Michoacan, Mexico)
Abstract:
We consider the Cauchy problem for the Schr\"{o}dinger
equation with a quadratic nonlinearity in one space dimension
$$
iu_{t}+\frac{1}{2}u_{xx}=t^{-\alpha}| u_x| ^2,\quad u(0,x) = u_0(x),
$$
where $\alpha \in (0,1)$. From the heuristic point of view,
solutions to this problem should have a quasilinear character
when $\alpha \in (1/2,1)$. We show in this paper that the
solutions do not have a quasilinear character for all
$\alpha \in (0,1)$
due to the special structure of the nonlinear term.
We also prove that for $\alpha \in [1/2,1)$ if the initial data
$u_0\in H^{3,0}\cap H^{2,2}$ are small, then
the solution has a slow time decay such as $t^{-\alpha /2}$.
For $\alpha \in (0,1/2)$, if we assume that the initial data
$u_0$ are
analytic and small, then the same time decay occurs.
Submitted May 22, 2001. Published July 25, 2001.
Math Subject Classifications: 35Q55, 74G10, 74G25.
Key Words: Schrodinger equation; large time behaviour;
quadratic nonlinearity.