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arXiv · 1704.00862

The Cauchy Problem for nonlinear Quadratic Interactions of the Schr\"odinger type in one dimensional space

Isnaldo Isaac
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Abstract

In this work I study the well-posedness of the Cauchy problem associated with the coupled Schr\"odinger equations {with quadratic nonlinearities}, which appears modeling problems in nonlinear optics. I obtain the local well-posedness for data {in Sobolev spaces} with low regularity. {To obtain} the local theory, I prove new bilinear estimates for the coupling terms of the system in the continuous case. Concerning global results, in the continuous case, I establish the global well-posedness in $H^s(\mathbb{R})\times H^s(\mathbb{R})$, for some negatives indexes $s$. The proof of the global result uses the \textbf{I}-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao.

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Isnaldo Isaac. 2017-04-04. The Cauchy Problem for nonlinear Quadratic Interactions of the Schr\"odinger type in one dimensional space. https://arxiv.org/abs/1704.00862

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