International Journal of Mathematics and Mathematical Sciences
Volume 29 (2002), Issue 9, Pages 501-516
doi:10.1155/S0161171202007652
Asymptotic expansion of small analytic solutions to the quadratic nonlinear Schrödinger equations in two-dimensional spaces
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.
Abstract
We study asymptotic behavior in time of global small solutions to
the quadratic nonlinear Schrödinger equation in
two-dimensional spaces i∂tu+(1/2)Δu=𝒩(u), (t,x)∈ℝ×ℝ2;u(0,x)=φ(x), x∈ℝ2, where 𝒩(u)=Σj,k=12(λjk(∂xju)(∂xku)+μjk(∂xju¯)(∂xku¯)), where
λjk,μjk∈ℂ. 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.