TY - JOUR

T1 - Asymptotic expansion of small analytic solutions to the quadratic NONLINEAR schrödinger equations in two-dimensional spaces

AU - Hayashi, Nakao

AU - Naumkin, Pavel I.

N1 - Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 2002

Y1 - 2002

N2 - 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.

AB - 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.

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U2 - 10.1155/S0161171202007652

DO - 10.1155/S0161171202007652

M3 - Article

AN - SCOPUS:17844386044

VL - 29

SP - 501

EP - 516

JO - International Journal of Mathematics and Mathematical Sciences

JF - International Journal of Mathematics and Mathematical Sciences

SN - 0161-1712

IS - 9

ER -