Kuruk, Gamze
(2019)
*Comparison of solutions of some pairs of nonlinear wave equations.*
[Thesis]

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## Abstract

In this thesis, we compare solutions of the Camassa-Holm equation with solutions of the Double Dispersion equation and the Hunter-Saxton equation. In the rst part of this thesis work, we determine a class of Boussinesq-type equations from which can be asymptotically derived. We use an expansion determined by two small positive parameters measuring nonlinear and dispersive e ects. We then rigorously show that solutions of the Camassa-Holm equation are well approximated by corresponding solutions of a certain class of the Double Dispersion equation over a long time scale. Finally we show that any solution of the Double Dispersion equation can be written as the sum of solutions of the two decoupled Camassa-Holm equations moving in opposite directions up to a small error. We observe that the approximation error for the decoupled problem is greater than the approximation error characterized by single Camassa-Holm approximation. We also obtain similar results for Benjamin-Bona-Mahony approximation to the Double Dispersion equation in the long wave limit. In the literature, Hunter- Saxton equation arises as high frequency limit of the Camassa-Holm equation. In the second part of this thesis work, we establish convergence results between the solutions of the Hunter-Saxton equations and the solutions of the Camassa-Holm equation in periodic setting providing a precise estimate for the approximation error.

Item Type: | Thesis |
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Uncontrolled Keywords: | Double dispersion equation. -- Camassa-Holm equation. -- Hunter-Saxton equation. -- Asymptotic expansion. -- Decoupling. -- Nonlinearity. -- Long wave. -- High frequency. -- Long time existence. -- Approximation error. -- İkili dispersif denklem. -- Camassa-Holm denklemi. -- Periyodik Hunter-Saxton denklemi. -- Asimptotik açılım. -- Doğusal olmama. -- Uzun dalga. -- Yüksek frekans. -- Uzun zamanda varlık. -- Yaklaşım hatası. -- Periyodik çözüm. |

Subjects: | Q Science > QA Mathematics |

Divisions: | Faculty of Engineering and Natural Sciences > Basic Sciences > Mathematics Faculty of Engineering and Natural Sciences |

Depositing User: | IC-Cataloging |

Date Deposited: | 21 Oct 2019 13:27 |

Last Modified: | 26 Apr 2022 10:32 |

URI: | https://research.sabanciuniv.edu/id/eprint/39354 |