Electron. J. Diff. Eqns., Vol. 2008(2008), No. 15, pp. 1-38.

Asymptotic behavior for a quadratic nonlinear Schrodinger equation

Nakao Hayashi, Pavel I. Naumkin

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 $u_{1}\in {H}^{2,2}$ we prove that there exists a unique global solution $u\in {C}([1,\infty );{H}^{2,2})$ 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 1, 2008.
Math Subject Classifications: 35B40, 35Q55.
Key Words: Nonlinear Schrodinger equation; large time asymptotic; self-similar solutions.

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Nakao Hayashi
Department of Mathematics, Graduate School of Science
Osaka University, Osaka
Toyonaka, 560-0043, Japan
email: nhayashi@math.wani.osaka-u.ac.jp
Pavel I. Naumkin
Instituto de Matemáticas
Universidad Nacional Autónoma de México
Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán, Mexico
email: pavelni@matmor.unam.mx

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