A Note on the Degasperis-Procesi Equation

Mustafa, Octavian G.

doi:10.2991/jnmp.2005.12.1.2

Cited by 68 publications

(49 citation statements)

References 20 publications

(19 reference statements)

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“…Lundmark and Szmigielski [15,16] used an inverse scattering approach to determine an explicit formula for the general npeakon solution of the Degasperis-Procesi equation (1.1). Mustafa [17] showed that smooth solutions to (1.1) have the infinite speed of propagation property.…”

Section: Introductionmentioning

confidence: 99%

Numerical schemes for computing discontinuous solutions of the Degasperis Procesi equation

Coclite

Karlsen

Risebro

2007

IMA Journal of Numerical Analysis

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Abstract. Recent work [4] has shown that the Degasperis-Procesi equation is well-posed in the class of (discontinuous) entropy solutions. In the present paper we construct numerical schemes and prove that they converge to entropy solutions. Additionally, we provide several numerical examples accentuating that discontinuous (shock) solutions form independently of the smoothness of the initial data. Our focus on discontinuous solutions contrasts notably with the existing literature on the Degasperis-Procesi equation, which seems to emphasize similarities with the Camassa-Holm equation (bi-Hamiltonian structure, integrabillity, peakon solutions, H 1 as the relevant functional space).

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Section: Introductionmentioning

confidence: 99%

Numerical schemes for computing discontinuous solutions of the Degasperis Procesi equation

Coclite

Karlsen

Risebro

2007

IMA Journal of Numerical Analysis

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“…Holm and Staley [20] studied stability of solitons and peakons of equation (1.5) numerically. Analogously to the case of the Camassa-Holm equation [5], Henry [18] and Mustafa [27] showed that smooth solutions to equation (1.5) have an infinite speed of propagation.…”

Section: Introductionmentioning

confidence: 99%

Stability of peakons for the Degasperis‐Procesi equation

Lin

Liu

2008

Comm Pure Appl Math

138

100

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The Degasperis-Procesi equation can be derived as a member of a one-parameter family of asymptotic shallow-water approximations to the Euler equations with the same asymptotic accuracy as that of the Camassa-Holm equation. In this paper, we study the orbital stability problem of the peaked solitons to the Degasperis-Procesi equation on the line. By constructing a Lyapunov function, we prove that the shapes of these peakon solitons are stable under small perturbations.

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“…Matsuno [42] studied multisoliton solutions and their peakon limits. Analogous to the case of the Camassa-Holm equation [10], Henry [32] and Mustafa [44] showed that smooth solutions to Eq. (1.2) have infinite speed of propagation.…”

Section: Introductionmentioning

confidence: 87%

Blow-Up Phenomena and Decay for the Periodic Degasperis-Procesi Equation with Weak Dissipation

Yin²

2008

JNMP

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In the paper, several problems on the periodic Degasperis-Procesi equation with weak dissipation are investigated. At first, the local well-posedness of the equation is established by Kato's theorem and a precise blow-up scenario of the solutions is given. Then, several criteria guaranteeing the blow-up of the solutions are presented. Moreover, the blow-up rate and blow-up set of the blowing-up solutions are discussed. Furthermore, it is proved that the equation has global solutions and these global solutions decay to zero as time goes to infinite provided the potentials associated to their initial date are of one sign.

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A Note on the Degasperis-Procesi Equation

Abstract: We prove that smooth solutions of the Degasperis-Procesi equation have infinite propagation speed: they loose instantly the property of having compact support.

Cited by 68 publications

References 20 publications

Numerical schemes for computing discontinuous solutions of the Degasperis Procesi equation

Numerical schemes for computing discontinuous solutions of the Degasperis Procesi equation

Stability of peakons for the Degasperis‐Procesi equation

Blow-Up Phenomena and Decay for the Periodic Degasperis-Procesi Equation with Weak Dissipation

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