An extension of integrable peakon equations with cubic nonlinearity

Geng, Xianguo; Xue, Bo

doi:10.1088/0951-7715/22/8/004

Cited by 151 publications

(117 citation statements)

References 18 publications

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“…This equation has a similar structure to CH and DP, but instead has cubic nonlinearities in the right hand side. We mention here also the Geng-Xue (GX) system [11] u xxt − u t = (u x − u xxx )uv + 3(u − u xx )vu x ,…”

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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Section: The Novikov Equationmentioning

confidence: 99%

New Phenomena in the World of Peaked Solitons

Kardell¹

2016

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“…However, it is difficult to associate the nonlinear evolution equations with corresponding spectral problems. Therefore, it is important for us to search for the new spectral problems and the relevant hierarchies of nonlinear evolution equations [14][15][16][17][18][19][20][21]. It is well-known that the trace identity is a powerful tool for constructing Hamiltonian structures of soliton equations, from which the Hamiltonian structures of many soliton equations are obtained [5,22].…”

Section: Introductionmentioning

confidence: 99%

A hierarchy of new nonlinear evolution equations and generalized bi-Hamiltonian structures

Wei

Geng

2015

Applied Mathematics and Computation

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“…Moreover, as a generalization of Eq. (1.1), Geng and Xue [9] studied multipeakon solutions for a coupled equation with cubic nonlinearity. As Li and Liu [10] stated, Eq.…”

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confidence: 99%

Further results on the smooth and nonsmooth solitons of the Novikov equation

Pan

2016

Nonlinear Dyn

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This paper is concerned with the smooth and nonsmooth soliton solutions of the Novikov equation based on the bifurcation method of dynamical systems. Two interesting results are highlighted. First, the new Hamiltonian function is established in the case of ϕ 2 < c while ϕ 2 > c is discussed in Li (Int J Bifurcat Chaos 24(3):1450037 2014). Second, we prove that the corresponding traveling wave system of the Novikov equation exists new smooth and nonsmooth soliton solutions.

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An extension of integrable peakon equations with cubic nonlinearity

Cited by 151 publications

References 18 publications

New Phenomena in the World of Peaked Solitons

New Phenomena in the World of Peaked Solitons

A hierarchy of new nonlinear evolution equations and generalized bi-Hamiltonian structures

Further results on the smooth and nonsmooth solitons of the Novikov equation

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