Stability of stationary solutions for nonintegrable peakon equations

Hone, Andrew N. W.; Lafortune, Stéphane

doi:10.1016/j.physd.2013.11.006

Cited by 13 publications

(9 citation statements)

References 42 publications

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“…The case of b = 2 is exactly the CH equation, while the case of b = 3 recovers the DP equation. According to various tests for integrability, it is known that the cases of b = 2 and b = 3 are the only integrable equations within this family [32][33][34][35]. However, for any b, all those equations admit peakon solutions [31].…”

Section: Introductionmentioning

confidence: 99%

A Synthetical Two‐Component Model with Peakon Solutions

2015

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A generalized two-component model with peakon solutions is proposed in this paper. It allows an arbitrary function to be involved in as well as including some existing integrable peakon equations as special reductions. The generalized two-component system is shown to possess Lax pair and infinitely many conservation laws. Bi-Hamiltonian structures and peakon interactions are discussed in detail for typical representative equations of the generalized system. In particular, a new type of N -peakon solution, which is not in the traveling wave type, is obtained from the generalized system.

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Section: Introductionmentioning

confidence: 99%

A Synthetical Two‐Component Model with Peakon Solutions

2015

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“…Nevertheless, for b 0.85 we noticed that peakons were emitted from the ramp-cliffs, with the former emerging as robust traveling waves. We investigated this byproduct of the numerical scheme by considering the implications of Theorem 3 in [20]. In particular, it can be shown that if m(x, t = 0) > 0, then m(x, t) > 0, ∀t > 0 holds which in fact is the case as per the Gaussian initial data employed in this work.…”

Section: Numerical Timesteppingmentioning

confidence: 94%

“…The behavior observed separately in each of the parameter ranges b > 1 and b < −1 can be understood as particular instances of the soliton resolution conjecture [19], a somewhat loosely defined conjecture which states that for suitable dispersive wave equations, solutions with "generic" initial data will decompose into a finite number of solitary waves plus a radiation part which disperses away. The authors of [20] provide a first step towards explaining this phenomenon analytically in the "lefton" regime b < −1. Indeed, they show that in this parameter range a single lefton solution is orbitally stable, by applying the approach of Grillakis, Shatah and Strauss in [21].…”

Section: Introductionmentioning

confidence: 99%

The Stability of the $b$-family of Peakon Equations

Charalampidis¹,

Parker²,

Kevrekidis³

et al. 2020

Preprint

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In the present work we revisit the b-family model of peakon equations, containing as special cases the b = 2 (Camassa-Holm) and b = 3 (Degasperis-Procesi) integrable examples. We establish information about the point spectrum of the peakon solutions and notably find that for suitably smooth perturbations there exists point spectrum in the right half plane rendering the peakons unstable for b < 1. We explore numerically these ideas in the realm of fixed-point iterations, spectral stability analysis and time-stepping of the model for the different parameter regimes. In particular, we identify exact, stationary (spectrally stable) lefton solutions for b < −1, and for −1 < b < 1, we dynamically identify ramp-cliff solutions as dominant states in this regime. We complement our analysis by examining the breakup of smooth initial data into stable peakons for b > 1. While many of the above dynamical features had been explored in earlier studies, in the present work, we supplement them, wherever possible, with spectral stability computations.

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“…[43]), which says that for any suitable dispersive evolutionary PDE (not necessarily integrable), generic initial data should decompose into a train of solitary waves together with radiation which decays to zero as t → ∞. Apart from intensive studies of the integrable cases b = 2, 3, further analytical and numerical support for the behaviour reported by Holm and Staley has taken a while to materialize: an orbital stability result for a single lefton when b < −1 was proved in [27], while there are various well-posedness and rigidity results (see [29,38] and references); yet linear stability/instability results for peakons, ramp/cliff solutions and leftons across the full range of b values have been found only very recently [7,31].…”

Section: Introductionmentioning

confidence: 97%

“…(See [27] for instance, where a Banach subspace of a weighted Sobolev space was considered in order to prove an orbital stability property of stationary solutions when b < −1. )…”

Section: Introductionmentioning

confidence: 99%

Similarity reductions of peakon equations: the $b$-family

Barnes,

Hone

2022

Preprint

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The b-family is a one-parameter family of Hamiltonian partial differential equations of non-evolutionary type, which arises in shallow water wave theory. It admits a variety of solutions, including the celebrated peakons, which are weak solutions in the form of peaked solitons with a discontinuous first derivative at the peaks, as well as other interesting solutions that have been obtained in exact form and/or numerically. In each of the special cases b = 2, 3 (the Camassa-Holm and Degasperis-Procesi equations, respectively) the equation is completely integrable, in the sense that it admits a Lax pair and an infinite hierarchy of commuting local symmetries, but for other values of the parameter b it is non-integrable. After a discussion of travelling waves via the use of a reciprocal transformation, which reduces to a hodograph transformation at the level of the ordinary differential equation satisfied by these solutions, we apply the same technique to the scaling similarity solutions of the b-family, and show that when b = 2 or 3 this similarity reduction is related by a hodograph transformation to particular cases of the Painlevé III equation, while for all other choices of b the resulting ordinary differential equation is not of Painlevé type.

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Stability of stationary solutions for nonintegrable peakon equations

Cited by 13 publications

References 42 publications

A Synthetical Two‐Component Model with Peakon Solutions

A Synthetical Two‐Component Model with Peakon Solutions

The Stability of the $b$-family of Peakon Equations

Similarity reductions of peakon equations: the $b$-family

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