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Forced Oscillations of the Korteweg-de Vries Equation and Their Stability

Usman, Muhammad
http://rave.ohiolink.edu/etdc/view?acc_num=ucin1186552805

Abstract Details

2007, PhD, University of Cincinnati, Arts and Sciences : Mathematical Sciences.
The equation of Korteweg and de Vries was derived as a model for propagation of surface water waves along the channel. This also approximates the models in nonlinear studies that includes and balance weak nonlinear and dispersive effects. In particular, the Korteweg-de Vries (KdV) equation is commonly accepted as a mathematical model for the unidirectional propagation of small-amplitude long waves in a nonlinear dispersive medium. Laboratory experiments have shown that when nonlinear, dispersive waves are forced periodically at one end of the channel, the signal eventually becomes temporally periodic at each spatial point. From dynamical system point of view the damped forced KdV equations have been studied by Ghidaglia and Sell and You. They have proved the existence of global attractor for the system. Zhang studied a damped forced KdV-equation posed on a finite domain with homogeneous Dirichlet boundary conditions and proved that if the external excitation is time periodic with the small amplitude, then the system admits a unique time periodic solution, which forms a inertial manifold for the infinite dimensional dynamical system. This work studies the Korteweg-de Vries equation posed on a bounded domain. It is shown that if the boundary forcing is time periodic with small amplitude then the corresponding solution of the system eventually becomes time-periodic. If the system is considered without the initial condition as an infinite dimensional dynamical system in the Hilbert space L2(0,1), it is also shown that for a given small amplitude periodic boundary forcing, the system admits a unique time periodic solution, that as a limit cycle, is locally exponentially stable.
Dr. Bingyu Zhang (Advisor)
103 p.
Mathematics
Forced oscillation; stability; the BBM equation; the KdV equation; time-periodic solution

Recommended Citations

Citations

  • Usman, M. (2007). Forced Oscillations of the Korteweg-de Vries Equation and Their Stability [Doctoral dissertation, University of Cincinnati]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1186552805

    APA Style (7th edition)

  • Usman, Muhammad. Forced Oscillations of the Korteweg-de Vries Equation and Their Stability. 2007. University of Cincinnati, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ucin1186552805.

    MLA Style (8th edition)

  • Usman, Muhammad. "Forced Oscillations of the Korteweg-de Vries Equation and Their Stability." Doctoral dissertation, University of Cincinnati, 2007. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1186552805

    Chicago Manual of Style (17th edition)

ucin1186552805
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© 2007, all rights reserved.
This open access ETD is published by University of Cincinnati and OhioLINK.