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On the generalized Benjamin-Ono equation. (English) Zbl 0804.35105
The initial-value problem for the generalized Benjamin-Ono (B-O) equation \[ \partial_ tu + u^ k \partial_ xu - \partial_ x D_ xu = 0,\;k \in \mathbb{Z}^ + \] is studied in Sobolev spaces \(H^ s(\mathbb{R})\). For small data and higher nonlinearities \((k \leq 2)\) new local and global results are established. The method the authors use with respect to the BO equation is quite general, and it allows to extend to other nonlinear models, e.g. to the generalized nonlinear Schrödinger equation of the form \(\partial_ tu - i \partial^ 2_ x u + P(u, \partial_ x u, u^*, \partial_ x u^*) = 0\).

MSC:
35Q30 Navier-Stokes equations
35G25 Initial value problems for nonlinear higher-order PDEs
35D99 Generalized solutions to partial differential equations
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