International Journal of Mathematics and Mathematical Sciences
Volume 29 (2002), Issue 9, Pages 501-516

Asymptotic expansion of small analytic solutions to the quadratic nonlinear Schrödinger equations in two-dimensional spaces

Nakao Hayashi1 and Pavel I. Naumkin2

1Department of Mathematics, Graduate School of Science, Osaka University, Osaka 560-0043, Japan
2Instituto de Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán, Mexico

Received 15 February 2001; Revised 20 May 2001

Copyright © 2002 Nakao Hayashi and Pavel I. Naumkin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study asymptotic behavior in time of global small solutions to the quadratic nonlinear Schrödinger equation in two-dimensional spaces itu+(1/2)Δu=𝒩(u), (t,x)×2;u(0,x)=φ(x), x2, where 𝒩(u)=Σj,k=12(λjk(xju)(xku)+μjk(xju¯)(xku¯)), where λjk,μjk. We prove that if the initial data φ satisfy some analyticity and smallness conditions in a suitable norm, then the solution of the above Cauchy problem has the asymptotic representation in the neighborhood of the scattering states.