A New Integrable Equation with Peakon Solutions

Abstract

We consider a new partial differential equation, of a similar form to the Camassa-Holm shallow water wave equation, which was recently obtained by Degasperis and Procesi using the method of asymptotic integrability. We prove the exact integrability of the new equation by constructing its Lax pair, and we explain its connection with a negative flow in the Kaup-Kupershmidt hierarchy via a reciprocal transformation. The infinite sequence of conserved quantities is derived together with a proposed bi-Hamiltonian structure. The equation admits exact solutions in the form of a superposition of multi-peakons, and we describe the integrable finite-dimensional peakon dynamics and compare it with the analogous results for Camassa-Holm peakons.

14 pages, 2 figures -- to appear in NEEDS 2001 Proceedings

Cited by in corpus (31)

• Wave Structures and Nonlinear Balances in a Family of 1+1 Evolutionary PDEs
• Multi-peakon solutions of the Degasperis-Procesi equation
• Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1+1 nonlinear evolutionary pde
• Inverse Scattering Transform for the Degasperis-Procesi Equation
• A family of wave-breaking equations generalizing the Camassa-Holm and Novikov equations
• The geometry of the two-component Camassa-Holm and Degasperis-Procesi equations
• Dressing Method for the Degasperis-Procesi Equation
• Hamiltonian Structures for the Ostrovsky-Vakhnenko Equation
• Well-posedness, travelling waves and geometrical aspects of generalizations of the Camassa-Holm equation
• Group Analysis of the Novikov Equation
• Degasperis-Procesi peakon dynamical system and finite Toda lattice of CKP type
• The periodic b-equation and Euler equations on the circle
• Stability of smooth solitary waves in the b-Camassa-Holm equation
• Local well-posedness of the EPDiff equation: a survey
• Multi loop soliton solutions and their interactions in the Degasperis-Procesi equation
• The inverse spectral problem for the discrete cubic string
• Existence, continuation, persistence and dynamics of solutions for a generalized 0-Holm-Staley equation
• An Application of Pfaffians to multipeakons of the Novikov equation and the finite Toda lattice of BKP type
• Stability in the energy space of the sum of N peakons for the Degasperis-Procesi equation
• A Remark on the Stability of Peakons for the Degasperis-Procesi Equation
• On weakly singular and fully nonlinear travelling shallow capillary-gravity waves in the critical regime
• The well-posedness, blow-up and travelling waves for a two-component Fornberg-Whitham system
• Self-Similar Blowup Solutions to the 2-Component Degasperis-Procesi Shallow Water System
• Global existence and blow-up for a weakly dissipative $\mu$DP equation
• High order finite difference WENO methods with unequal-sized sub-stencils for the Degasperis-Procesi type equations
• Generalised Fourier Transform and Perturbations to Soliton Equations
• NLS-type equations from quadratic pencil of Lax operators: negative flows
• The Cauchy problem and continuation of periodic solutions for a generalized Camassa-Holm equation
• Weak solution for Stochastic Degasperis-Procesi Equation
• Unique continuation results for abstract quasi-linear evolution equations in Banach spaces
• A rigidity result for the Holm-Staley b-family of equations with application to the asymptotic stability of the Degasperis-Procesi peakon