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Local well‐posedness for a system of quadratic nonlinear Schrödinger equations in one or two dimensions
Author(s) -
Zhang Huali
Publication year - 2016
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3863
Subject(s) - mathematics , semigroup , initial value problem , space (punctuation) , nonlinear system , limit (mathematics) , quadratic equation , mathematical analysis , function (biology) , measure (data warehouse) , dirac (video compression format) , cauchy problem , mathematical physics , quantum mechanics , physics , geometry , philosophy , linguistics , database , evolutionary biology , computer science , neutrino , biology
In this article, the local well‐posedness of Cauchy's problem is explored for a system of quadratic nonlinear Schrödinger equations in the space L p ( R n ). In a special case of mass resonant 2 × 2 system, it is well known that this problem is well posed in H s ( s ≥0) and ill posed in H s ( s < 0) in two‐space dimensions. By translation on a linear semigroup, we show that the general system becomes locally well posed in L p ( R 2 ) for 1 < p < 2, for which p can arbitrarily be close to the scaling limit p c =1. In one‐dimensional case, we show that the problem is locally well posed in L 1 ( R ); moreover, it has a measure valued solution if the initial data are a Dirac function. Copyright © 2016 John Wiley & Sons, Ltd.