Asymptotic expansion of small analytic solutions to the quadratic nonlinear Schrödinger equations in two-dimensional spaces

Nakao Hayashi and Pavel I. Naumkin

International Journal of Mathematics and Mathematical Sciences, 2002, vol. 29, 1-16

Abstract:

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 ) Δ u = 𝒩 ( u ) , ( t , x ) ∈ ℠× â„ 2 ; u ( 0 , x ) = φ ( x ) , x ∈ â„ 2 , where 𝒩 ( u ) = Σ j , k = 1 2 ( λ j k ( ∂ x j u ) ( ∂ x k u ) + μ j k ( ∂ x j u ¯ ) ( ∂ x k u ¯ ) ) , where λ j k , μ j k ∈ â„‚ . 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.

Date: 2002
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:319879

DOI: 10.1155/S0161171202007652

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