liu.seSearch for publications in DiVA
Change search
Refine search result
1 - 19 of 19
CiteExportLink to result list
Permanent link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Rows per page
  • 5
  • 10
  • 20
  • 50
  • 100
  • 250
Sort
  • Standard (Relevance)
  • Author A-Ö
  • Author Ö-A
  • Title A-Ö
  • Title Ö-A
  • Publication type A-Ö
  • Publication type Ö-A
  • Issued (Oldest first)
  • Issued (Newest first)
  • Created (Oldest first)
  • Created (Newest first)
  • Last updated (Oldest first)
  • Last updated (Newest first)
  • Disputation date (earliest first)
  • Disputation date (latest first)
  • Standard (Relevance)
  • Author A-Ö
  • Author Ö-A
  • Title A-Ö
  • Title Ö-A
  • Publication type A-Ö
  • Publication type Ö-A
  • Issued (Oldest first)
  • Issued (Newest first)
  • Created (Oldest first)
  • Created (Newest first)
  • Last updated (Oldest first)
  • Last updated (Newest first)
  • Disputation date (earliest first)
  • Disputation date (latest first)
Select
The maximal number of hits you can export is 250. When you want to export more records please use the Create feeds function.
  • 1.
    Gomez, Daniel
    et al.
    University of Saskatchewan, Canada .
    Lundmark, Hans
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Szmigielski, Jacek
    University of Saskatchewan, Canada .
    The Canada Day Theorem2013In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 20, no 1Article in journal (Refereed)
    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.

  • 2.
    Hone, Andrew N W
    et al.
    University of Kent.
    Lundmark, Hans
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Szmigielski, Jacek
    University of Saskatchewan.
    Explicit multipeakon solutions of Novikovs cubically nonlinear integrable Camassa-Holm type equation2009In: DYNAMICS OF PARTIAL DIFFERENTIAL EQUATIONS, ISSN 1548-159X, Vol. 6, no 3, p. 253-289Article in journal (Refereed)
    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.

  • 3.
    Kardell, Marcus
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Lundmark, Hans
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Peakon-antipeakon solutions of the Novikov equationManuscript (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.

  • 4. Kohlenberg, Jennifer
    et al.
    Lundmark, Hans
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Szmigielski, Jacek
    The inverse spectral problem for the discrete cubic string2007In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 23, no 1, p. 99-121Article in journal (Refereed)
    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.

  • 5.
    Lundmark, Hans
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    A new class of integrable Newton systems2001In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 8, p. 195-199Article in journal (Refereed)
    Abstract [en]

    A new class of integrable Newton systems in R-n is presented. They are characterized by the existence of two quadratic integrals of motion of so-called cofactor type, and are therefore called cofactor pair systems. This class includes as special cases conservative systems separable in elliptic or parabolic coordinates, as well as many Newton systems previously derived as reductions of soliton hierarchies.

  • 6.
    Lundmark, Hans
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Degasperis-Procesi peakons and the discrete `cubic string'2004In: Orthogonal Polynomials; Interdisciplinary Aspects,2004, 2004Conference paper (Other academic)
  • 7.
    Lundmark, Hans
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Direct and inverse spectral problem for a non-selfadjonint third order generalization of the discrete string equation2004In: Colloquium on Operator Theory on the occasion of the retirement of Heinz Langer,2004, 2004Conference paper (Refereed)
  • 8.
    Lundmark, Hans
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Driven Newton equations and separable time-dependent potentials2004In: State-of-the-arts of classical separability theory for differential equations,2004, 2004Conference paper (Other academic)
  • 9.
    Lundmark, Hans
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Formation and dynamics of shock waves in the Degasperis-Procesi equation2007In: Journal of nonlinear science, ISSN 0938-8974, E-ISSN 1432-1467, Vol. 17, no 3, p. 169-198Article in journal (Refereed)
    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.

  • 10.
    Lundmark, Hans
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Higher-Dimensional Integrable Newton Systems with Quadratic Integrals of Motion2003In: Studies in applied mathematics (Cambridge), ISSN 0022-2526, E-ISSN 1467-9590, Vol. 1103, no 3, p. 257-296Article in journal (Refereed)
  • 11.
    Lundmark, Hans
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Peakons and shockpeakons: an introduction to the world of nonsmooth solitons2008In: Banach Lie-Poisson spaces and integrable systems,2008, 2008Conference paper (Other academic)
  • 12.
    Lundmark, Hans
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Peakons and shockpeakons in the Degasperis-Procesi equation2007In: NEEDS 2007 Nonlinear Evolution Equations and Dynamical Systems,2007, 2007Conference paper (Other academic)
  • 13.
    Lundmark, Hans
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Some recent developments in the study of peaked solitons2008In: Réunion Painlevé,2008, 2008Conference paper (Other academic)
  • 14.
    Lundmark, Hans
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Shuaib, Budor
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Ghostpeakons and Characteristic Curves for the Camassa-Holm, Degasperis-Procesi and Novikov Equations2019In: SIGMA. Symmetry, Integrability and Geometry, ISSN 1815-0659, E-ISSN 1815-0659, Vol. 15, article id 017Article in journal (Refereed)
    Abstract [en]

    We derive explicit formulas for the characteristic curves associated with the multipeakon solutions of the Camassa-Holm, Degasperis-Procesi and Novikov equations. Such a curve traces the path of a fluid particle whose instantaneous velocity equals the elevation of the wave at that point (or the square of the elevation, in the Novikov case). The peakons themselves follow characteristic curves, and the remaining characteristic curves can be viewed as paths of "ghostpeakons" with zero amplitude; hence, they can be obtained as solutions of the ODEs governing the dynamics of multipeakon solutions. The previously known solution formulas for multipeakons only cover the case when all amplitudes are nonzero, since they are based upon inverse spectral methods unable to detect the ghostpeakons. We show how to overcome this problem by taking a suitable limit in terms of spectral data, in order to force a selected peakon amplitude to become zero. Moreover, we use direct integration to compute the characteristic curves for the solution of the Degasperis-Procesi equation where a shockpeakon forms at a peakon-antipeakon collision. In addition to the theoretical interest in knowing the characteristic curves, they are also useful for plotting multipeakon solutions, as we illustrate in several examples.

  • 15.
    Lundmark, Hans
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Szmigielski, Jacek
    University of Saskatchewan.
    An inverse spectral problem related to the Geng–Xue two-component peakon equation2016In: Memoirs of the American Mathematical Society, ISSN 0065-9266, E-ISSN 1947-6221, Vol. 244, no 1155Article in journal (Refereed)
    Abstract [en]

    We solve a spectral and an inverse spectral problem arising in the computation of peakon solutions to the two-component PDE derived by Geng and Xue as a generalization of the Novikov and Degasperis-Procesi equations. Like the spectral problems for those equations, this one is of a ‘discrete cubic string’ type – a nonselfadjoint generalization of a classical inhomogeneous string – but presents some interesting novel features: there are two Lax pairs, both of which contribute to the correct complete spectral data, and the solution to the inverse problem can be expressed using quantities related to Cauchy biorthogonal polynomials with two different spectral measures. The latter extends the range of previous applications of Cauchy biorthogonal polynomials to peakons, which featured either two identical, or two closely related, measures. The method used to solve the spectral problem hinges on the hidden presence of oscillatory kernels of Gantmacher-Krein type implying that the spectrum of the boundary value problem is positive and simple. The inverse spectral problem is solved by a method which generalizes, to a nonselfadjoint case, M. G. Krein’s solution of the inverse problem for the Stieltjes string.

  • 16.
    Lundmark, Hans
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Szmigielski, Jacek
    Continuous and Discontinuous Piecewise Linear Solutions of the Linearly Forced Inviscid Burgers Equation2008In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 15, p. 264-276Article in journal (Refereed)
    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.

  • 17.
    Lundmark, Hans
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Szmigielski, Jacek
    University of Saskatchewan.
    Degasperis-Procesi peakons and the discrete cubic string2005In: International Mathematics Research Papers, ISSN 1687-3017, E-ISSN 1687-3009, Vol. 2005, p. 53-116Article in journal (Refereed)
    Abstract [en]

    [No abstract available]

  • 18.
    Lundmark, Hans
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Szmigielski, Jacek
    Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, Saskatchewan, Canada.
    Dynamics of interlacing peakons (and shockpeakons) in the Geng–Xue equation2017In: Journal of Integrable Systems, ISSN 2058-5985, Vol. 2, no 1, article id xyw014Article in journal (Refereed)
    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.

  • 19.
    Lundmark, Hans
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Szmigielski, Jacek
    Multi-peakon solutions of the Degasperis-Procesi equation2003In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 19, no 6, p. 1241-1245Article in journal (Refereed)
    Abstract [en]

    We present an inverse scattering approach for computing n-peakon solutions of the Degasperis-Procesi equation (a modification of the Camassa-Holm (CH) shallow water equation). The associated non-self-adjoint spectral problem is shown to be amenable to analysis using the isospectral deformations induced from the n-peakon solution, and the inverse problem is solved by a method generalizing the continued fraction solution of the peakon sector of the CH equation.

1 - 19 of 19
CiteExportLink to result list
Permanent link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf