An Inverse Spectral Problem Related to the Geng–Xue Two-Component Peakon Equation

Lundmark, Hans; Szmigielski, Jacek

doi:10.1090/memo/1155

Cited by 44 publications

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“…Pure multipeakon solutions to Novikov and GX were studied in [13] and [21] respectively, while the peakon-antipeakon interactions are the topic of Paper II.…”

Section: The Novikov Equationmentioning

confidence: 99%

New Phenomena in the World of Peaked Solitons

Kardell¹

2016

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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 "New solutions with peakon creation in the Camassa-Holm and Novikov 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 valid weak solutions to the CamassaHolm 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.i ii Populärvetenskaplig sammanfattningInom vågteori studeras så kallade solitoner, vilka kan beskrivas som vågpaket som rör sig med konstant form och hastighet. Typiska egenskaper är att utbredningen i rummet är begränsad, samt att två solitoner som kolliderar kan passera genom varandra utan att ändra form.Fenomenet beskrevs redan 1834 av John Scott Russell, som ridande längs en kanal följde en "rundad, slät, väldefinierad upphöjning av vatten, vilken fortsatte sin bana längs kanalen synbarligen utan att ändra form eller förlora fart". Dåvarande våglära kunde inte förklara uppkomsten av sådana vågor, men moderna hydrodynamiska teorier innehåller ett antal modeller där solitoner är ett naturligt koncept.I denna avhandling studeras vågekvationer som tillåter en särskild typ av spetsiga ...

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“…Pure multipeakon solutions to Novikov and GX were studied in [13] and [21] respectively, while the peakon-antipeakon interactions are the topic of Paper II.…”

Section: The Novikov Equationmentioning

confidence: 99%

New Phenomena in the World of Peaked Solitons

Kardell¹

2016

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“…The mixed case was studied in [2] and [21] for CH and DP respectively. Pure multipeakon solutions to Novikov and GX were studied in [12] and [18] respectively.…”

Section: X(t) M(t)mentioning

confidence: 99%

Contributions to the theory of peaked solitons

Kardell¹

2014

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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 GengXue equations, and new peakon-like unbounded solutions to the Camassa-Holm, DegasperisProcesi 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 → +∞ and/or x → −∞. 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". i ii Populärvetenskaplig sammanfattningInom vågteori studeras så kallade solitoner, vilka kan beskrivas som vågpaket som rör sig med konstant form och hastighet. Typiska egenskaperär att utbredningen i rummetär begränsad, samt att två solitoner som kolliderar kan passera genom varandra utan attändra form.Fenomenet beskrevs redan 1834 av John Scott Russell, som ridande längs en kanal följde en "rundad, slät, väldefinierad upphöjning av vatten, vilken fortsatte sin bana längs kanalen synbarligen utan attändra form eller förlora fart". Dåvarande våglära kunde inte förklara uppkomsten av sådana vågor, men moderna hydrodynamiska teorier innehåller ett antal modeller där solitonerär ett naturligt koncept.I denna avhandling studeras vågekvationer som tillåter en särskild typ av spetsiga solitoner, så kallade peakoner (från engelskans 'peaked soliton'). Avhandlingen utgörs av två artiklar som på olika sätt bidrar till grundförståelsen av detta f...

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“…In this system, peakons in u are not allowed to occupy the same sites as peakons in v. The interlacing peakon configuration (with first a peakon in u, then a peakon in v, then a peakon in u, and so on) has been solved by Lundmark and Szmigielski [16,17], by the inverse spectral transform method. Our contribution is to find the solution for and arbitrary peakon configuration in the Geng-Xue equation using ideas that originated during the study of ghostpeakons in Paper I.…”

Section: Overview Of Paper IImentioning

confidence: 99%

“…where the positions and amplitudes depend on time as described by the explicit formulas found by Lundmark and Szmigielski [16]. In terms of the notation from .…”

Section: Overview Of Paper IImentioning

confidence: 99%

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Ghostpeakons

Shuaib¹

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In this thesis we study peakons (peaked solitons), a class of solutions which occur in certain wave equations, such as the Camassa-Holm shallow water equation and its mathematical relatives, the Degasperis-Procesi, Novikov and Geng-Xue equations. These four non-linear partial differential equations are all integrable systems in the sense of having Lax pairs, infinitely many conservation laws, and multipeakon solutions given by explicitly known formulas.In the first paper, we develop a method which uses so-called ghostpeakons (peakons with amplitude zero) to find explicit formulas for the characteristic curves associated with the multipeakon solutions of the Camassa-Holm, Degasperis-Procesi and Novikov equations.In the second paper, we use the ghostpeakon method to derive explicit formulas for arbitrary multipeakon solutions of the two-component Geng-Xue equation. The general case involves many inequivalent peakons configurations, depending on the order in which the peakons occur in the two components of the solution, and previously the solution was known only in the so-called interlacing case where the peakons lie alternatingly in one component and in the other. To obtain the solution formulas for an arbitrary configuration, we introduce auxiliary peakons to make the configuration interlacing. By taking suitable limits, we then drive the amplitudes of the auxiliary peakons to zero, leaving the solution formulas for the remaining ordinary peakons.Sammanfattning på svenska I denna avhandling studeras peakoner (från engelskans peakons, en förkortning för peaked solitons, dvs. spetsiga solitoner). Detta är en klass av lösningar som fö-rekommer i vissa vågekvationer, t.ex. Camassa-Holm-ekvationen från teorin för vågor i grunt vatten, samt de matematiskt närbesläktade Degasperis-Procesi-, Novikov-och Geng-Xue-ekvationerna. Samtliga dessa fyra ickelinjära partiella differentialekvationer är integrabla system i den meningen att de har Laxpar, oändligt många konserveringslagar, samt multipeakonlösningar som ges av explicita formler.I den första artikeln utvecklar vi en metod som använder s.k. spökpeakoner (peakoner med amplituden noll) för att finna explicita formler för de karaktäris-tiska kurvorna som hör till multipeakonlösningarna till Camassa-Holm-, Degasperis-Procesi-och Novikov-ekvationerna.I den andra artikeln använder vi spökpeakonmetoden för att härleda explicita formler för godtyckliga multipeakonlösningar till Geng-Xue-ekvationen, som är en tvåkomponentsekvation. Det allmänna fallet innefattar många icke-ekvivalenta peakonkonfigurationer, beroende på i vilken ordning peakonerna kommer i de två komponenterna i lösningen, och tidigare var lösningen känd enbart i det s.k. sammanflätade fallet där peakonerna ligger växelvis i den ena och den andra komponenten. För att erhålla lösningsformlerna för en godtycklig konfiguration inför vi hjälppeakoner som gör konfigurationen sammanflätad. Genom lämpliga gränsövergångar tvingar vi sedan amplituden för dessa hjälppeakoner att gå mot noll, och kvar blir då lösni...

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An Inverse Spectral Problem Related to the Geng–Xue Two-Component Peakon Equation

Cited by 44 publications

References 41 publications

New Phenomena in the World of Peaked Solitons

New Phenomena in the World of Peaked Solitons

Contributions to the theory of peaked solitons

Ghostpeakons

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