Geodesic
Asymptotic behavior for a quadratic nonlinear Schrödinger equation
Electronic Journal of Differential Equations, Tome 2008 (2008).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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},
     publisher = {mathdoc},
     volume = {2008},
     year = {2008},
     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
PB  - mathdoc
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
%I mathdoc
%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/