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Lundmark, H. & Szmigielski, J. (2017). Dynamics of interlacing peakons (and shockpeakons) in the Geng–Xue equation. Journal of Integrable Systems, 2(1), Article ID xyw014.
Open this publication in new window or tab >>Dynamics of interlacing peakons (and shockpeakons) in the Geng–Xue equation
2017 (English)In: Journal of Integrable Systems, ISSN 2058-5985, Vol. 2, no 1, article id xyw014Article in journal (Refereed) Published
Abstract [en]

We consider multipeakon solutions, and to some extent also multishockpeakon solutions, of a coupled two- component integrable PDE found by Geng and Xue as a generalization of Novikov’s cubically nonlinear Camassa–Holm type equation. In order to make sense of such solutions, we find it necessary to assume that there are no overlaps, meaning that a peakon or shockpeakon in one component is not allowed to occupy the same position as a peakon or shockpeakon in the other component. Therefore one can distinguish many inequivalent configurations, depending on the order in which the peakons or shockpeakons in the two components appear relative to each other. Here we are particularly interested in the case of interlacing peakon solutions, where the peakons alternatingly occur in one component and in the other. Based on explicit expressions for these solutions in terms of elementary functions, we describe the general features of the dynamics, and in particular the asymptotic large-time behaviour (assuming that there are no antipeakons, so that the solutions are globally defined). As far as the positions are concerned, interlacing Geng–Xue peakons display the usual scattering phenomenon where the peakons asymptotically travel with constant velocities, which are all distinct, except that the two fastest peakons (the fastest one in each component) will have the same velocity. However, in contrast to many other peakon equations, the amplitudes of the peakons will not in general tend to constant values; instead they grow or decay exponentially. Thus the logarithms of the amplitudes (as functions of time) will asymptotically behave like straight lines, and comparing these lines for large positive and negative times, one observes phase shifts similar to those seen for the positions of the peakons (and also for the positions of solitons in many other contexts). In addition to these K+K interlacing pure peakon solutions, we also investigate 1+1 shockpeakon solutions, and collisions leading to shock formation in a 2+2 peakon–antipeakon solution.

Place, publisher, year, edition, pages
Oxford University Press, 2017
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-139590 (URN)10.1093/integr/xyw014 (DOI)
Available from: 2017-08-09 Created: 2017-08-09 Last updated: 2017-08-30Bibliographically approved
Gomez, D., Lundmark, H. & Szmigielski, J. (2013). The Canada Day Theorem. The Electronic Journal of Combinatorics, 20(1)
Open this publication in new window or tab >>The Canada Day Theorem
2013 (English)In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 20, no 1Article in journal (Refereed) Published
Abstract [en]

The Canada Day Theorem is an identity involving sums of k x k minors of an arbitrary n x n symmetric matrix. It was discovered as a by-product of the work on so-called peakon solutions of an integrable nonlinear partial differential equation proposed by V. Novikov. Here we present another proof of this theorem, which explains the underlying mechanism in terms of the orbits of a certain abelian group action on the set of all k-edge matchings of the complete bipartite graph K-n,K-n.

Place, publisher, year, edition, pages
Electronic Journal of Combinatorics, 2013
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-89808 (URN)000314575000003 ()
Note

Funding Agencies|NSERC (Natural Sciences and Engineering Research Council of Canada) USRA|414987|Swedish Research Council (Vetenskapsradet)|VR 2010-5822||NSERC 163953|

Available from: 2013-03-07 Created: 2013-03-07 Last updated: 2017-12-06
Hone, A. N., Lundmark, H. & Szmigielski, J. (2009). Explicit multipeakon solutions of Novikovs cubically nonlinear integrable Camassa-Holm type equation. DYNAMICS OF PARTIAL DIFFERENTIAL EQUATIONS, 6(3), 253-289
Open this publication in new window or tab >>Explicit multipeakon solutions of Novikovs cubically nonlinear integrable Camassa-Holm type equation
2009 (English)In: DYNAMICS OF PARTIAL DIFFERENTIAL EQUATIONS, ISSN 1548-159X, Vol. 6, no 3, p. 253-289Article in journal (Refereed) Published
Abstract [en]

Recently Vladimir Novikov found a new integrable analogue of the Camassa-Holm equation which has nonlinear terms that are cubic, rather than quadratic, and which admits peaked soliton solutions (peakons). In this paper, the explicit formulas for multipeakon solutions of Novikovs cubically nonlinear equation are calculated, using the matrix Lax pair found by Hone and Wang. By a transformation of Liouville type, the associated spectral problem is related to a cubic string equation, which is dual to the cubic string that was previously found in the work of Lundmark and Szmigielski on the multipeakons of the Degasperis-Procesi equation.

Keywords
Peakons, cubic string, Novikovs equation, Degasperis-Procesi equation, distributional Lax pair, sum of minors
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-51597 (URN)
Available from: 2009-11-09 Created: 2009-11-09 Last updated: 2013-11-14
Lundmark, H. & Szmigielski, J. (2008). Continuous and Discontinuous Piecewise Linear Solutions of the Linearly Forced Inviscid Burgers Equation. Journal of Nonlinear Mathematical Physics, 15, 264-276
Open this publication in new window or tab >>Continuous and Discontinuous Piecewise Linear Solutions of the Linearly Forced Inviscid Burgers Equation
2008 (English)In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 15, p. 264-276Article in journal (Refereed) Published
Abstract [en]

We study a class of piecewise linear solutions to the inviscid Burgers equation driven by a linear forcing term. Inspired by the analogy with peakons, we think of these solutions as being made up of solitons situated at the breakpoints. We derive and solve ODEs governing the soliton dynamics, first for continuous solutions, and then for more general shock wave solutions with discontinuities. We show that triple collisions of solitons cannot take place for continuous solutions, but give an example of a triple collision in the presence of a shock.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-17147 (URN)10.2991/jnmp.2008.15.s3.26 (DOI)
Available from: 2009-03-07 Created: 2009-03-07 Last updated: 2017-12-13
Lundmark, H. (2008). Peakons and shockpeakons: an introduction to the world of nonsmooth solitons. In: Banach Lie-Poisson spaces and integrable systems,2008.
Open this publication in new window or tab >>Peakons and shockpeakons: an introduction to the world of nonsmooth solitons
2008 (English)In: Banach Lie-Poisson spaces and integrable systems,2008, 2008Conference paper, Published paper (Other academic)
Keywords
matematik, differentialekvationer
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-43841 (URN)74940 (Local ID)74940 (Archive number)74940 (OAI)
Available from: 2009-10-10 Created: 2009-10-10 Last updated: 2013-11-14
Lundmark, H. (2008). Some recent developments in the study of peaked solitons. In: Réunion Painlevé,2008.
Open this publication in new window or tab >>Some recent developments in the study of peaked solitons
2008 (English)In: Réunion Painlevé,2008, 2008Conference paper, Published paper (Other academic)
Keywords
matematik, differentialekvationer
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-43804 (URN)74846 (Local ID)74846 (Archive number)74846 (OAI)
Available from: 2009-10-10 Created: 2009-10-10 Last updated: 2013-11-14
Lundmark, H. (2007). Formation and dynamics of shock waves in the Degasperis-Procesi equation. Journal of nonlinear science, 17(3), 169-198
Open this publication in new window or tab >>Formation and dynamics of shock waves in the Degasperis-Procesi equation
2007 (English)In: Journal of nonlinear science, ISSN 0938-8974, E-ISSN 1432-1467, Vol. 17, no 3, p. 169-198Article in journal (Refereed) Published
Abstract [en]

Solutions of the Degasperis-Procesi nonlinear wave equation may develop discontinuities in finite time. As shown by Coclite and Karlsen, there is a uniquely determined entropy weak solution which provides a natural continuation of the solution past such a point. Here we study this phenomenon in detail for solutions involving interacting peakons and antipeakons. We show that a jump discontinuity forms when a peakon collides with an antipeakon, and that the entropy weak solution in this case is described by a "shockpeakon" ansatz reducing the PDE to a system of ODEs for positions, momenta, and shock strengths.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-39443 (URN)10.1007/s00332-006-0803-3 (DOI)48399 (Local ID)48399 (Archive number)48399 (OAI)
Available from: 2009-10-10 Created: 2009-10-10 Last updated: 2017-12-13
Lundmark, H. (2007). Peakons and shockpeakons in the Degasperis-Procesi equation. In: NEEDS 2007 Nonlinear Evolution Equations and Dynamical Systems,2007.
Open this publication in new window or tab >>Peakons and shockpeakons in the Degasperis-Procesi equation
2007 (English)In: NEEDS 2007 Nonlinear Evolution Equations and Dynamical Systems,2007, 2007Conference paper, Published paper (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-40102 (URN)52263 (Local ID)52263 (Archive number)52263 (OAI)
Available from: 2009-10-10 Created: 2009-10-10 Last updated: 2013-11-14
Kohlenberg, J., Lundmark, H. & Szmigielski, J. (2007). The inverse spectral problem for the discrete cubic string. Inverse Problems, 23(1), 99-121
Open this publication in new window or tab >>The inverse spectral problem for the discrete cubic string
2007 (English)In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 23, no 1, p. 99-121Article in journal (Refereed) Published
Abstract [en]

Given a measure m on the real line or a finite interval, the cubic string is the third-order ODE - ′′′ ≤ zm where z is a spectral parameter. If equipped with Dirichlet-like boundary conditions this is a non-self-adjoint boundary value problem which has recently been shown to have a connection to the Degasperis-Procesi nonlinear water wave equation. In this paper, we study the spectral and inverse spectral problem for the case of Neumann-like boundary conditions which appear in a high-frequency limit of the Degasperis-Procesi equation. We solve the spectral and inverse spectral problem for the case of m being a finite positive discrete measure. In particular, explicit determinantal formulae for the measure m are given. These formulae generalize Stieltjes' formulae used by Krein in his study of the corresponding second-order ODE -″ ≤ zm. © 2007 IOP Publishing Ltd.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-39442 (URN)10.1088/0266-5611/23/1/005 (DOI)48398 (Local ID)48398 (Archive number)48398 (OAI)
Available from: 2009-10-10 Created: 2009-10-10 Last updated: 2017-12-13
Lundmark, H. & Szmigielski, J. (2005). Degasperis-Procesi peakons and the discrete cubic string. International Mathematics Research Papers, 2005, 53-116
Open this publication in new window or tab >>Degasperis-Procesi peakons and the discrete cubic string
2005 (English)In: International Mathematics Research Papers, ISSN 1687-3017, E-ISSN 1687-3009, Vol. 2005, p. 53-116Article in journal (Refereed) Published
Abstract [en]

[No abstract available]

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-24643 (URN)6829 (Local ID)6829 (Archive number)6829 (OAI)
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2017-12-13
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0003-4137-8272

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