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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the generalized Benjamin-Ono equation


Authors: Carlos E. Kenig, Gustavo Ponce and Luis Vega
Journal: Trans. Amer. Math. Soc. 342 (1994), 155-172
MSC: Primary 35Q53; Secondary 35Q55
MathSciNet review: 1153015
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Abstract: We study well-posedness of the initial value problem for the generalized Benjamin-Ono equation $ {\partial _t}u + {u^k}{\partial _x}u - {\partial _x}{D_x}u = 0$, $ k \in {\mathbb{Z}^ + }$, in Sobolev spaces $ {H^s}(\mathbb{R})$. For small data and higher nonlinearities $ (k \geq 2)$ new local and global (including scattering) results are established. Our method of proof is quite general. It combines several estimates concerning the associated linear problem with the contraction principle. Hence it applies to other dispersive models. In particular, it allows us to extend the results for the generalized Benjamin-Ono to nonlinear Schrödinger equations (or systems) of the form $ {\partial _t}u - i\partial _x^2u + P(u,{\partial _x}u,\bar u,{\partial _x}\bar u) = 0$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1153015-4
PII: S 0002-9947(1994)1153015-4
Keywords: Generalized Benjamin-Ono equation, initial value problem, well-posedness
Article copyright: © Copyright 1994 American Mathematical Society