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Contributions to the theory of peaked solitons
Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
2014 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

The aim of this work is to present some new contributions to the theory of peaked solitons. The thesis contains two papers, named "Lie symmetry analysis of the Novikov and Geng-Xue equations, and new peakon-like unbounded solutions to the Camassa-Holm, Degasperis-Procesi and Novikov equations'' and "Peakon-antipeakon solutions of the Novikov equation'' respectively.

In the first paper, a new kind of peakon-like solutions to the Novikov equation is obtained, by transforming the one-peakon solution via a Lie symmetry transformation. This new kind of solution is unbounded as x tends to infinity and/or minus infinity. It also has a peak, though only for some interval of time. We make sure that the peakon-like function is still a solution in the weak sense for those times where the function is non-differentiable. We find that similar solutions, with peaks living only for some interval in time, are valid weak solutions to the Camassa-Holm equation, though these can not be obtained via a symmetry transformation.

The second paper covers peakon-antipeakon solutions of the Novikov equation, on the basis of known solution formulas from the pure peakon case. A priori, these formulas are valid only for some interval of time and only for some initial values. The aim of the article is to study the Novikov multipeakon solution formulas in detail, to overcome these problems. We find that the formulas for locations and heights of the peakons are valid for all times at least in an ODE sense. Also, we suggest a procedure of how to deal with multipeakons where the initial conditions are such that the usual spectral data are not well-defined as residues of single poles of a Weyl function. In particular we cover the interaction between one peakon and one antipeakon, revealing some unexpected properties. For example, with complex spectral data, the solution is shown to be periodic, except for a translation, with an infinite number of collisions between the peakon and the antipeakon. Also, plotting solution formulas for larger number of peakons shows that there are similarities to the phenomenon called "waltzing peakons''.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2014. , 8 p.
Series
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1650
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-105710DOI: 10.3384/lic.diva-105710ISBN: 978-91-7519-373-1 (print)OAI: oai:DiVA.org:liu-105710DiVA: diva2:709792
Presentation
2014-04-02, Alan Turing, E-huset, Campus Valla, Linköpings universitet, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2014-04-09 Created: 2014-04-03 Last updated: 2014-04-09Bibliographically approved
List of papers
1. Lie symmetry analysis of the Novikov and Geng-Xue equations, and new peakon-like solutions to the Camassa-Holm, Degasperis-Procesi and Novikov equations
Open this publication in new window or tab >>Lie symmetry analysis of the Novikov and Geng-Xue equations, and new peakon-like solutions to the Camassa-Holm, Degasperis-Procesi and Novikov equations
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this article we use the Maple package Jets to compute symmetry groups of the Novikov equation and the Geng-Xue system. The group generators correspond to transformations that take solutions of the equations to other solutions. By applying one of the transformations for the Novikov equation to the known one-peakon solution, we find a new kind of unbounded solutions with peakon creation, i.e., these functions are smooth solutions for some time interval, then after a certain finite time, a peak is created. We show that the functions are still weak solutions to the Novikov equation for those times where the peak lives. We also find similar unbounded solutions with peakon creation in the related Camassa-Holm equation, by making an ansatz inspired by the Novikov solutions. Finally, we see that the same ansatz for the Degasperis-Procesi equation yields unbounded solutions where a peakon is present for all times.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-105708 (URN)
Available from: 2014-04-03 Created: 2014-04-03 Last updated: 2014-04-09
2. Peakon-antipeakon solutions of the Novikov equation
Open this publication in new window or tab >>Peakon-antipeakon solutions of the Novikov equation
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Certain nonlinear partial differential equations admit multisoliton solutions in the form of a superposition of peaked waves, so-called peakons. The Camassa–Holm andDegasperis–Procesi equations are twowellknown examples, and a more recent one is the Novikov equation, which has cubic nonlinear terms instead of quadratic. In this article we investigate multipeakon solutions of theNovikov equation, in particular interactions between peakons with positive amplitude and antipeakons with negative amplitude. The solutions are given by explicit formulas, which makes it possible to analyze them in great detail. As in the Camassa–Holm case, the slope of the wave develops a singularity when a peakon collides with an antipeakon, while the wave itself remains continuous and can be continued past the collision to provide a global weak solution. However, the Novikov equation differs in several interesting ways from other peakon equations, especially regarding asymptotics for large times. For example, peakons and antipeakons both travel to the right, making it possible for several peakons and antipeakons to travel together with the same speed and collide infinitely many times. Such clusters may exhibit very intricate periodic or quasi-periodic interactions. It is also possible for peakons to have the same asymptotic velocity but separate at a logarithmic rate; this phenomenon is associated with coinciding eigenvalues in the spectral problem coming from the Lax pair, and requires nontrivial modifications to the previously known solution formulas which assume that all eigenvalues are simple. To facilitate the reader’s understanding of these multipeakon phenomena, we have included a particularly detailed description of the case with just one peakon and one antipeakon, and also made an effort to provide plenty of graphics for illustration.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-105709 (URN)
Available from: 2014-04-03 Created: 2014-04-03 Last updated: 2016-01-26Bibliographically approved

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Kardell, Marcus

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