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Transactions of the American Mathematical Society

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



Large time asymptotics of solutions
to the generalized Benjamin-Ono equation

Authors: Nakao Hayashi and Pavel I. Naumkin
Journal: Trans. Amer. Math. Soc. 351 (1999), 109-130
MSC (1991): Primary 35Q55
MathSciNet review: 1491867
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Abstract: We study the asymptotic behavior for large time of solutions to the Cauchy problem for the generalized Benjamin-Ono (BO) equation: $u_{t} + (|u|^{\rho -1}u)_{x} + \mathcal{H} u_{xx} = 0 $, where $\mathcal{H}$ is the Hilbert transform, $x, t \in {\mathbf{R}}$, when the initial data are small enough. If the power $\rho $ of the nonlinearity is greater than $3$, then the solution of the Cauchy problem has a quasilinear asymptotic behavior for large time. In the case $\rho=3$ critical for the asymptotic behavior i.e. for the modified Benjamin-Ono equation, we prove that the solutions have the same $L^{\infty }$ time decay as in the corresponding linear BO equation. Also we find the asymptotics for large time of the solutions of the Cauchy problem for the BO equation in the critical and noncritical cases. For the critical case, we prove the existence of modified scattering states if the initial function is sufficiently small in certain weighted Sobolev spaces. These modified scattering states differ from the free scattering states by a rapidly oscillating factor.

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Additional Information

Nakao Hayashi
Affiliation: Department of Applied Mathematics, Science University of Tokyo, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162, Japan

Pavel I. Naumkin
Affiliation: Instituto de Fisica y Matematica, Universidad Michoacana, AP 2-82, CP 58040, Morelia, Michoacana, Mexico

Received by editor(s): August 9, 1996
Article copyright: © Copyright 1999 American Mathematical Society

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