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Sharp well-posedness results for the Schrödinger-Benjamin-Ono system. (English) Zbl 1342.35329
The author studies the coupled Schrödinger-Benjamin-Ono system \[ \begin{aligned} i\partial_{t}u+\partial_{x}^{2}u & =\alpha u v, \\ \partial_{t}v+\nu\mathcal{H}\partial_{x}^{2}v & =\beta\partial_{x}(|u|^{2}).\end{aligned} \] Here, \(\mathcal{H}\) is the Hilbert transform, and the independent variable \(x\) is taken to be in \(\mathbb{R}\). The non-resonant case is the case in which \(|\nu|\neq 1\). Prior existence results have been established, showing well-posedness of this system in the Sobolev spaces \(H^{s}\times H^{s-1/2}\).
Here, the author studies well-posedness in \(H^{s}\times H^{s'}\), without requiring \(s'=s-1/2.\) Under certain conditions on \(s\), \(s'\), local well-posedness is proved. Furthermore, in both the resonant and non-resonant cases, for a variety of values of \(s\), \(s'\), the failure of the solution map to be \(C^{2}\) is proved.

35Q55 NLS equations (nonlinear Schrödinger equations)
35G25 Initial value problems for nonlinear higher-order PDEs
Full Text: Euclid arXiv