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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On the generalized Benjamin-Ono equation
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by Carlos E. Kenig, Gustavo Ponce and Luis Vega PDF
Trans. Amer. Math. Soc. 342 (1994), 155-172 Request permission

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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Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 342 (1994), 155-172
  • MSC: Primary 35Q53; Secondary 35Q55
  • DOI: https://doi.org/10.1090/S0002-9947-1994-1153015-4
  • MathSciNet review: 1153015