Asymptotic behavior for a quadratic nonlinear Schrodinger equation

Abstract

We study the initial-value problem for the quadratic nonlinear Schrodinger equation $$displaylines{ iu_{t}+frac{1}{2}u_{xx}=partial _{x}overline{u}^{2},quad xin mathbb{R},; t>1, cr u(1,x)=u_{1}(x),quad xin mathbb{R}. }$$ For small initial data $u_{1}in mathbf{H}^{2,2}$ we prove that there exists a unique global solution $uin mathbf{C}([1,infty );mathbf{H}^{2,2})$ of this Cauchy problem. Moreover we show that the large time asymptotic behavior of the solution is defined in the region $|x|leq Csqrt{t}$ 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.

Keywords

• Nonlinear Schrodinger equation
• large time asymptotic
• self-similar solutions