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New Phenomena in the World of Peaked Solitons
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The aim of this work is to present new contributions to the theory of peaked solitons. The thesis consists of two papers,which are named “Newsolutionswith peakon creation in the Camassa–HolmandNovikov equations” and “Peakon-antipeakon solutions of the Novikov equation” respectively.

In Paper I, a new kind of peakon-like solution to the Novikov equation is discovered, by transforming the one-peakon solution via a Lie symmetry transformation. This new kind of solution is unbounded as x → +∞ and/or x → –∞, and has a peak, though only for some interval of time. Thus, the solutions exhibit creation and/or destruction of peaks. 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 validweak solutions to the Camassa–Holm equation, though it appears that these can not be obtained via a symmetry transformation.

In Paper II we investigate multipeakon solutions of the Novikov 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.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2016. , 14 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1737
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-124307DOI: 10.3384/diss.diva-124307ISBN: 9789176858431 (print)OAI: oai:DiVA.org:liu-124307DiVA: diva2:897568
Public defence
2016-02-26, Visionen, B-huset, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2016-01-26 Created: 2016-01-26 Last updated: 2017-12-15Bibliographically approved
List of papers
1. New solutions with peakon creation in the Camassa-Holm and Novikov equations
Open this publication in new window or tab >>New solutions with peakon creation in the Camassa-Holm and Novikov equations
2015 (English)In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 22, no 1Article in journal (Refereed) Published
Abstract [en]

In this article we study a new kind of unbounded solutions to the Novikov equation, found via a Lie symmetry analysis. These solutions exhibit peakon creation, i.e., these solutions are smooth up until a certain finite time, at which a peak is created. We show that the functions are still weak solutions 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.

Place, publisher, year, edition, pages
Taylor and Francis: STM, Behavioural Science and Public Health Titles, 2015
Keyword
Novikov; Camassa-Holm; peakon; weak solution; 76M60; 58D19; 35D30
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-113491 (URN)10.1080/14029251.2015.996435 (DOI)000346180500001 ()
Available from: 2015-01-19 Created: 2015-01-19 Last updated: 2017-12-05
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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