Local well‐posedness for a system of quadratic nonlinear Schrödinger equations in one or two dimensions |
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| Authors: | Huali Zhang |
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| Affiliation: | School of Mathematical Sciences, Fudan University, Shanghai, China |
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| Abstract: | In this article, the local well‐posedness of Cauchy's problem is explored for a system of quadratic nonlinear Schrödinger equations in the space Lp( R n). In a special case of mass resonant 2 × 2 system, it is well known that this problem is well posed in Hs(s≥0) and ill posed in Hs(s < 0) in two‐space dimensions. By translation on a linear semigroup, we show that the general system becomes locally well posed in Lp( R 2) for 1 < p < 2, for which p can arbitrarily be close to the scaling limit pc=1. In one‐dimensional case, we show that the problem is locally well posed in L1( R ); moreover, it has a measure valued solution if the initial data are a Dirac function. Copyright © 2016 John Wiley & Sons, Ltd. |
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| Keywords: | nonlinear Schrö dinger equations local well‐posedness |
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