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Smooth and Non-Smooth Traveling Wave Solutions of Some Generalized Camassa-Holm Equations

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Date Issued:
2013
Abstract/Description:
In this thesis we employ two recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of recently derived integrable family of generalized Camassa-Holm (GCH) equations. In the first part, a novel application of phase-plane analysis is employed to analyze the singular traveling wave equations of four GCH equations, i.e. the possible non-smooth peakon, cuspon and compacton solutions. Two of the GCH equations do no support singular traveling waves. We generalize an existing theorem to establish the existence of peakon solutions of the third GCH equation. This equation is found to also support four segmented, non-smooth M-wave solutions. While the fourth supports both solitary (peakon) and periodic (cuspon) cusp waves in different parameter regimes.In the second part of the thesis, smooth traveling waves of the four GCH equations are considered. Here, we use a recent technique to derive convergent multi-infinite series solutions for the homoclinic and heteroclinic orbits of their traveling-wave equations, corresponding to pulse and front (kink or shock) solutions respectively of the original PDEs. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. Of course, the convergence rate is not comparable to typical asymptotic series. However, asymptotic solutions for global behavior along a full homoclinic/heteroclinic orbit are currently not available.
Title: Smooth and Non-Smooth Traveling Wave Solutions of Some Generalized Camassa-Holm Equations.
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Name(s): Rehman, Taslima, Author
Choudhury, Sudipto, Committee Chair
Nevai, Andrew, Committee Member
Rollins, David, Committee Member
, Committee Member
University of Central Florida, Degree Grantor
Type of Resource: text
Date Issued: 2013
Publisher: University of Central Florida
Language(s): English
Abstract/Description: In this thesis we employ two recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of recently derived integrable family of generalized Camassa-Holm (GCH) equations. In the first part, a novel application of phase-plane analysis is employed to analyze the singular traveling wave equations of four GCH equations, i.e. the possible non-smooth peakon, cuspon and compacton solutions. Two of the GCH equations do no support singular traveling waves. We generalize an existing theorem to establish the existence of peakon solutions of the third GCH equation. This equation is found to also support four segmented, non-smooth M-wave solutions. While the fourth supports both solitary (peakon) and periodic (cuspon) cusp waves in different parameter regimes.In the second part of the thesis, smooth traveling waves of the four GCH equations are considered. Here, we use a recent technique to derive convergent multi-infinite series solutions for the homoclinic and heteroclinic orbits of their traveling-wave equations, corresponding to pulse and front (kink or shock) solutions respectively of the original PDEs. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. Of course, the convergence rate is not comparable to typical asymptotic series. However, asymptotic solutions for global behavior along a full homoclinic/heteroclinic orbit are currently not available.
Identifier: CFE0004918 (IID), ucf:49637 (fedora)
Note(s): 2013-08-01
M.S.
Sciences, Mathematics
Masters
This record was generated from author submitted information.
Subject(s): Camassa-Holm Equations -- Homoclinic orbits -- Heteroclinic orbits -- Reversible and non-reversible systems -- Traveling wave solutions -- Peakons -- Cuspons
Persistent Link to This Record: http://purl.flvc.org/ucf/fd/CFE0004918
Restrictions on Access: public 2013-08-15
Host Institution: UCF

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