In this paper, we prove an orbital stability result for the Degasperis-Procesi peakon with respect to perturbations having a momentum density that is first negative and then positive. This leads to ...the orbital stability of the antipeakon-peakon profile with respect to such perturbations.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
The aim of this paper is to prove that the Degasperis–Procesi antipeakon–peakon profile is asymptotically stable for all time. We start by proving the asymptotic stability of a single ...Degasperis–Procesi peakon and antipeakon with respect to perturbations having a momentum density that is first negative and then positive. Then this result is extended towards a well-ordered trains of antipeakons–peakons under such perturbations. In particular, the asymptotic stability of the antipeakon–peakon profile holds.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
We consider the non-local formulation of the Degasperis-Procesi equation ut+uux+L(32u2)x=0, where L is the non-local Fourier multiplier operator with symbol m(ξ)=(1+ξ2)−1. We show that all L∞, ...pointwise travelling-wave solutions are bounded above by the wave-speed and that if the maximal height is achieved they are peaked at those points, otherwise they are smooth. For sufficiently small periods we find the highest, peaked, travelling-wave solution as the limiting case at the end of the main bifurcation curve of P-periodic solutions. The results imply that there are no L∞ travelling cuspon solutions to the Degasperis-Procesi equation.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Recently Vladimir Novikov found a new integrable analogue of the Camassa-Holm equation which has nonlinear terms that are cubic, rather than quadratic, and which admits peaked soliton solutions ...(peakons). In this paper, the explicit formulas for multipeakon solutions of Novikovs cubically nonlinear equation are calculated, using the matrix Lax pair found by Hone and Wang. By a transformation of Liouville type, the associated spectral problem is related to a cubic string equation, which is dual to the cubic string that was previously found in the work of Lundmark and Szmigielski on the multipeakons of the Degasperis-Procesi equation.
In this Letter, we present a bi-Hamiltonian structure for the two-component Novikov equation. We also show that proper reduction of this bi-Hamiltonian structure leads to the Hamiltonian operators ...found by Hone and Wang for the Novikov equation.
► A bi-Hamiltonian structure is constructed for a two-component Novikov equation. ► The complete proof is given for Jacobi properties and compatibilty of the relevant operators. ► A reduction is considred for the Hamiltonian structures.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK
The Degasperis-Procesi equation is an approximating model of shallow-water wave propagating mainly in one direction to the Euler equations. Such a model equation is analogous to the Camassa-Holm ...approximation of the two-dimensional incompressible and irrotational Euler equations with the same asymptotic accuracy, and is completely integrable with the bi-Hamiltonian structure. In the present study, we establish existence and spectral stability results of localized smooth solitons to the Degasperis-Procesi equation on the real line. The stability proof relies essentially on refined spectral analysis of the linear operator corresponding to the second-order variational derivative of the local Hamiltonian of the Degasperis-Procesi equation.
L'équation de Degasperis-Procesi est un modèle approximatif d'ondes en eau basse se propageant principalement dans une direction vers les équations d'Euler. Une telle équation de modèle est analogue à l'approximation de Camassa-Holm des équations d'Euler incompressibles en deux-dimentions et irrotationnelles, avec la même précision asymptotique, et ellle est intégrable avec la structure bi-hamiltonienne. Dans l'étude présente, nous établissons les résultats d'existence et de stabilité spectrale de solitons lisses et localisées à l'équation de Degasperis-Procesi sur la droite réelle. La preuve de stabilité repose essentiellement sur une analyse spectrale raffinée de l'opérateur linéaire correspondant à la dérivée variationnelle du second ordre de l'hamiltonien de l'équation de Degasperis-Procesi.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
In this paper, we investigate the initial value problem of the generalized Degasperis-Procesi equation. We establish some new local-in-space blow-up criterion. Our results extend the corresponding ...results in the previous literature.
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BFBNIB, GIS, IJS, KISLJ, NUK, PNG, UL, UM, UPUK
We classify all weak traveling wave solutions of the Degasperis–Procesi equation. In addition to smooth and peaked solutions, the equation is shown to admit more exotic traveling waves such as ...cuspons, stumpons, and composite waves.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
We consider the numerical integration of the Degasperis–Procesi equation, which was recently introduced as a completely integrable shallow water equation. For the equation, we propose nonlinear and ...linear finite difference schemes that preserve two invariants associated with the bi-Hamiltonian form of the equation at the same time. We also prove the unique solvability of the schemes, and show some numerical examples.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
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