Forced oscillations of a damped‎ ‎Korteweg-de Vries equation on a periodic domain

Document Type: Research Paper

Author

School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, ‎P‎. ‎R‎. ‎China.

Abstract

‎In this paper‎, ‎we investigate a damped Korteweg-de‎ ‎Vries equation with forcing on a periodic domain‎ ‎$\mathbb{T}=\mathbb{R}/(2\pi\mathbb{Z})$‎. ‎We can obtain that if the‎ ‎forcing is periodic with small amplitude‎, ‎then the solution becomes‎ ‎eventually time-periodic.

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Main Subjects


J. L. Bona, S. M. Sun and B. Y. Zhang, Forced oscillations of a damped Korteweg-deVries equation in a quarter plane, Commun. Contemp. Math. 5 (2003), no. 3, 369--400.

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to non-linear evolution equations, part II: The KdV equation, Geom. Funct. Anal. 3 (1993), no. 3, 209--262.

W. Craig and C. E. Wayne, Newton's method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math. 46 (1993), no. 11, 1409--1498.

C. Laurent, L. Rosier and B.Y. Zhang, Control and Stabilization of the Korteweg-deVries Equation on a Periodic Domain, Comm. Partial Differential Equations 35 (2010), no. 4, 707--744.

P. H. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations, Comm. Pure Appl. Math. 20 (1967) 145--205.

D. L. Russell and B. Y. Zhang, Controllability and stabilizability of the third order linear dispersion equation on a periodic domain, SIAM J. Control Optim. 31 (1993), no. 3, 659--676.

D. L. Russell and B. Y. Zhang, Exact controllability and stabilizability of the Korteweg-de Vries equation, Trans. Amer. Math. Soc. 348 (1996), no. 9, 3643--3672.

M. Usman and B. Y. Zhang, Forced oscillations of a class of nonlinear dispersive wave equations and their stability, J. Syst. Sci. Complex. 20 (2007), no. 2, 284--292.

C. E.Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys. 127 (1990), no. 3, 479--528.

B. Y. Zhang, Forced oscillation of the Korteweg-de Vries-Burgers equation and its stability, Control of Nonlinear Distributed Parameter Systems (College Station, TX, 1999), 337--357, Lecture Notes in Pure and Appl. Math., 218, Dekker, New York, 2001.


Volume 42, Issue 5
September and October 2016
Pages 1027-1038
  • Receive Date: 08 October 2014
  • Revise Date: 14 June 2015
  • Accept Date: 19 June 2015