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.

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.

UR - http://www.scopus.com/inward/record.url?scp=17844386044&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=17844386044&partnerID=8YFLogxK

U2 - 10.1155/S0161171202007652

DO - 10.1155/S0161171202007652

M3 - Article

AN - SCOPUS:17844386044

SN - 0161-1712

VL - 29

SP - 501

EP - 516

JO - International Journal of Mathematics and Mathematical Sciences

JF - International Journal of Mathematics and Mathematical Sciences

IS - 9

ER -