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Comparison of solutions of some pairs of nonlinear wave equations

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

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Official URL: http://risc01.sabanciuniv.edu/record=b2325807 (Table of contents)

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
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
ID Code:39354
Deposited By:IC-Cataloging
Deposited On:21 Oct 2019 13:27
Last Modified:15 Jul 2020 00:01

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