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

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

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.