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.