A view of the peakon world through the lens of approximation theory

We consider the scaling similarity solutions of two integrable cubically nonlinear partial diﬀerential equations (PDEs) that admit peaked soliton (peakon) solutions, namely the modiﬁed Camassa-Holm (mCH) equation and Novikov’s equation. By making use of suitable reciprocal transformations, which map the mCH equation and Novikov’s equation to a negative mKdV ﬂow and a negative Sawada-Kotera ﬂow, respectively, we show that each of these scaling similarity reductions is related via a hodograph transformation to an equation of Painlevé type: for the mCH equation, its reduction is of second order and second degree, while for Novikov’s equation the reduction is a particular case of Painlevé V. Furthermore, we show that each of these two diﬀerent Painlevé-type equations is related to the particular cases of Painlevé III that arise from analogous similarity reductions of the Camassa-Holm and the Degasperis-Procesi equation, respectively. For each of the cubically nonlinear PDEs considered, we also give explicit parametric forms of their periodic travelling wave solutions in terms of elliptic functions. We present some parametric plots of the latter, and, by using explicit algebraic solutions of Painlevé III, we do the same for some of the simplest examples of scaling similarity solutions, together with descriptions of their leading order asymptotic behaviour.

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“…. , N. For more details of peakons and their connections with approximation theory, see the recent review [31] and references therein.…”

Section: Background and Motivationmentioning

confidence: 99%

Similarity reductions of peakon equations: integrable cubic equations

Barnes¹,

Hone²,

Senthilvelan³

et al. 2022

J. Phys. A: Math. Theor.

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“…It was shown in [3,21,23] that the DP and CH equations describe the horizontal velocity u = u(t, x) for the unidirectional propagation of waves of a shallow water flowing over a flat bed at a certain depth. A review of applicability of these model equations as approximations of peaked waves in fluids was recently given in [33].…”

Section: Introductionmentioning

confidence: 99%

Stability of smooth periodic traveling waves in the Degasperis-Procesi equation

Geyer¹,

Pelinovsky²

2022

Preprint

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We derive a precise energy stability criterion for smooth periodic waves in the Degasperis-Procesi (DP) equation. Compared to the Camassa-Holm (CH) equation, the number of negative eigenvalues of an associated Hessian operator changes in the existence region of smooth perodic waves. We utilize properties of the period function with respect to two parameters in order to obtain a smooth existence curve for the family of smooth periodic waves of a fixed period. The energy stability condition is derived on parts of this existence curve which correspond to either one or two negative eigenvalues of the Hessian operator. We show numerically that the energy stability condition is satisfied on either part of the curve and prove analytically that it holds in a neighborhood of the boundary of the existence region of smooth periodic waves.

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“…The inverse scattering approach is a powerful tool to study the CH equation and analyze its dynamics [24]. The inverse scattering transform (IST) problem of the CH equation is a complicated issue [25].…”

mentioning

confidence: 99%

Ill-Posedness of a Three-Component Novikov System in Besov Spaces

Yu,

Zhou

2024

Mathematics

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In this paper, we consider the Cauchy problem for a three-component Novikov system on the line. We give a construction of the initial data (ρ0,u0,v0)∈Bp,∞σ−1(R)×Bp,∞σ(R)×Bp,∞σ(R) with σ>max3+1p,72,1≤p≤∞, such that the corresponding solution to the three-component Novikov system starting from (ρ0,u0,v0) is discontinuous at t=0 in the metric of Bp,∞σ−1(R)×Bp,∞σ(R)×Bp,∞σ(R), which implies the ill-posedness for this system in Bp,∞σ−1(R)×Bp,∞σ(R)×Bp,∞σ(R).

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A view of the peakon world through the lens of approximation theory

Cited by 14 publications

References 284 publications

Similarity reductions of peakon equations: integrable cubic equations

Similarity reductions of peakon equations: integrable cubic equations

Stability of smooth periodic traveling waves in the Degasperis-Procesi equation

Ill-Posedness of a Three-Component Novikov System in Besov Spaces

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