Geodesic
Asymptotic behavior for a quadratic nonlinear Schrödinger equation
Electronic journal of differential equations, Tome 2008 (2008)
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

$ \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.
Classification : 35B40, 35Q55
Keywords: nonlinear Schrödinger equation, large time asymptotic, self-similar solutions 
@article{EJDE_2008__2008__a3,
     author = {Hayashi,  Nakao and Naumkin,  Pavel I.},
     title = {Asymptotic behavior for a quadratic nonlinear {Schr\"odinger} equation},
     journal = {Electronic journal of differential equations},
     year = {2008},
     volume = {2008},
     zbl = {1136.35324},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2008__2008__a3/}
}
TY  - JOUR
AU  - Hayashi,  Nakao
AU  - Naumkin,  Pavel I.
TI  - Asymptotic behavior for a quadratic nonlinear Schrödinger equation
JO  - Electronic journal of differential equations
PY  - 2008
VL  - 2008
UR  - http://geodesic.mathdoc.fr/item/EJDE_2008__2008__a3/
LA  - en
ID  - EJDE_2008__2008__a3
ER  - 
%0 Journal Article
%A Hayashi,  Nakao
%A Naumkin,  Pavel I.
%T Asymptotic behavior for a quadratic nonlinear Schrödinger equation
%J Electronic journal of differential equations
%D 2008
%V 2008
%U http://geodesic.mathdoc.fr/item/EJDE_2008__2008__a3/
%G en
%F EJDE_2008__2008__a3
Hayashi,  Nakao; Naumkin,  Pavel I. Asymptotic behavior for a quadratic nonlinear Schrödinger equation. Electronic journal of differential equations, Tome 2008 (2008). http://geodesic.mathdoc.fr/item/EJDE_2008__2008__a3/