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Peakon-antipeakon solutions of the Novikov equation
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0003-4137-8272
(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: urn:nbn:se:liu:diva-105709OAI: oai:DiVA.org:liu-105709DiVA: diva2:709780
Available from: 2014-04-03 Created: 2014-04-03 Last updated: 2016-01-26Bibliographically approved
In thesis
1. Contributions to the theory of peaked solitons
Open this publication in new window or tab >>Contributions to the theory of peaked solitons
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:nbn:se:liu:diva-105710 (URN)10.3384/lic.diva-105710 (DOI)978-91-7519-373-1 (ISBN)
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
2. New Phenomena in the World of Peaked Solitons
Open this publication in new window or tab >>New Phenomena in the World of Peaked Solitons
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:nbn:se:liu:diva-124307 (URN)10.3384/diss.diva-124307 (DOI)9789176858431 (ISBN)
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

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