Asymptotic expansion of small analytic solutions to the quadratic NONLINEAR schrödinger equations in two-dimensional spaces

We study asymptotic behavior in time of global small solutions to the quadratic nonlinear Schrödinger equation in two-dimensional spaces i∂t u+ (1/2) Δu = 𝒩 (u), (t, x)∈ 2;u (0, x)=φ (x), x∈ 2, where 𝒩 (u) = ∑ j, k = 1 2 (λ jk (∂ xj u) (∂ xk u) + μ jk (∂ xj u⁻) (∂ xk u^-∼)), where λ jk,μ jk∈. 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.