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.