Global existence of small solutions to the quadratic nonlinear Schrödinger equations in two space dimensions

Nakao Hayashi; Pavel I. Naumkin

doi:10.1137/S0036141000372532

Global existence of small solutions to the quadratic nonlinear Schrödinger equations in two space dimensions

Nakao Hayashi, Pavel I. Naumkin

Research output: Contribution to journal › Article › peer-review

5 Citations (Scopus)

Abstract

We study a global existence in time of small solutions to the quadratic nonlinear Schrödinger equation in two space dimensions, (formula presented) where (formula presented) λ_jk, μ_jk ∈ C. We prove that if the initial data u₀ satisfy some analyticity and smallness conditions in a suitable norm, then the solution of the Cauchy problem (0.1) exists globally in time. Furthermore we prove the existence of the usual scattering states.

Original language	English
Pages (from-to)	1390-1403
Number of pages	14
Journal	SIAM Journal on Mathematical Analysis
Volume	32
Issue number	6
DOIs	https://doi.org/10.1137/S0036141000372532
Publication status	Published - 2001 Feb 1
Externally published	Yes

Keywords

Global existence
Nonlinear Schrödinger equations
Quadratic nonlinearities
Two spatial dimensions

ASJC Scopus subject areas

Analysis
Computational Mathematics
Applied Mathematics

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10.1137/S0036141000372532

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abstract = "We study a global existence in time of small solutions to the quadratic nonlinear Schr{\"o}dinger equation in two space dimensions, (formula presented) where (formula presented) λjk, μjk ∈ C. We prove that if the initial data u0 satisfy some analyticity and smallness conditions in a suitable norm, then the solution of the Cauchy problem (0.1) exists globally in time. Furthermore we prove the existence of the usual scattering states.",

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AU - Naumkin, Pavel I.

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N2 - We study a global existence in time of small solutions to the quadratic nonlinear Schrödinger equation in two space dimensions, (formula presented) where (formula presented) λjk, μjk ∈ C. We prove that if the initial data u0 satisfy some analyticity and smallness conditions in a suitable norm, then the solution of the Cauchy problem (0.1) exists globally in time. Furthermore we prove the existence of the usual scattering states.

AB - We study a global existence in time of small solutions to the quadratic nonlinear Schrödinger equation in two space dimensions, (formula presented) where (formula presented) λjk, μjk ∈ C. We prove that if the initial data u0 satisfy some analyticity and smallness conditions in a suitable norm, then the solution of the Cauchy problem (0.1) exists globally in time. Furthermore we prove the existence of the usual scattering states.

KW - Global existence

KW - Nonlinear Schrödinger equations

KW - Quadratic nonlinearities

KW - Two spatial dimensions

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JF - SIAM Journal on Mathematical Analysis

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ER -

Global existence of small solutions to the quadratic nonlinear Schrödinger equations in two space dimensions

Abstract

Keywords

ASJC Scopus subject areas

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