Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 15, pp. 1-38.

Title: Asymptotic behavior for a quadratic nonlinear
       Schrodinger equation

Authors: Nakao Hayashi    (Osaka Univ., Osaka, Japan)
         Pavel I. Naumkin (Univ. Nacional Autonoma, Mexico)

Abstract: 
 We study the initial-value problem for the quadratic nonlinear
 Schrodinger equation
 $$\displaylines{
 iu_{t}+\frac{1}{2}u_{xx}=\partial _{x}\overline{u}^{2},\quad x\in
 \mathbb{R},\; t>1, \cr
 u(1,x)=u_{1}(x),\quad x\in \mathbb{R}.
 }$$
 For small initial data $u_{1}\in \mathbf{H}^{2,2}$ we prove
 that there exists a unique global solution
 $u\in \mathbf{C}([1,\infty );\mathbf{H}^{2,2})$ of this Cauchy
 problem. Moreover we show that the large time asymptotic behavior
 of the solution  is defined in the region
 $|x|\leq C\sqrt{t}$ by the self-similar solution
 $\frac{1}{\sqrt{t}}MS(\frac{x}{\sqrt{t}})$ such that the total mass
 $$
 \frac{1}{\sqrt{t}}\int_{\mathbb{R}}MS(\frac{x}{\sqrt{t}})
 dx=\int_{\mathbb{R}}u_{1}(x)dx,
 $$
 and in the far region $|x|>\sqrt{t}$ the asymptotic  behavior of
 solutions has rapidly oscillating structure similar to that of the cubic
 nonlinear Schrodinger equations.
  
Submitted March 19, 2007. Published February 01, 2008.
Math Subject Classifications: 35B40, 35Q55.
Key Words: Nonlinear Schrodinger equation; large time asymptotic; 
           self-similar solutions.