A New Integrable Equation with Peakon Solutions

By Antonio Degasperis, Darryl D. Holm, Andrew N. W. Hone. 2002

774 citations recorded by the source corpus.

We consider a new partial differential equation recently obtained by Degasperis and Procesi using the method of asymptotic integrability; this equation has a form similar to the Camassa–Holm shallow water wave equation. We prove the exact integrability of the new equation by constructing its Lax pair and explain its relation to 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 as a superposition of multipeakons, and we describe the integrable finite-dimensional peakon dynamics and compare it with the analogous results for Camassa–Holm peakons.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/606140a1981675698893e683c6cff85431ff1da9

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Camassa–Holm, Korteweg–de Vries-5 and other asymptotically equivalent equations for shallow water waves (2003, methodology): “=3 in (4.4) recovers the DP equation of Degasperis and Procesi (1999), which was shown to be integrable in Degasperis et al. (2002).”
Well-posedness and global existence for the Novikov equation (2012, background): “Degasperis, Holm and Hone [9] prove the formal integrability of Equation (1.”
Autonomous Ovsyannikov theorem and applications to nonlocal evolution equations and systems (2016, methodology): “[23] A. Degasperis, D. Holm and A. Hone, A new integral equation with peakon solutions, Theoret.”
THE GEOMETRY OF THE TWO-COMPONENT CAMASSA-HOLM AND DEGASPERIS-PROCESI EQUATIONS (2010, background): “Both the CH and DP equations are integrable and admit peakon solutions [2, 6].”
Cuspons and Smooth Solitons of the Degasperis–Procesi Equation Under Inhomogeneous Boundary Condition (2007, background): “The DP equation has Lax pair [7] (therefore is integrable), and is able to be extended to a whole integrable hierarchy of equations with parametric solutions under some constraints [13, 17].”
Equations with Peakon Traveling Wave Solutions (2013, background): “For related results about them, we refer the reader to [2, 5–15, 17–19, 21] and the references therein.”
Periodic and solitary-wave solutions of the Degasperis-Procesi equation (2004, background): “More recently Degasperis, Holm and Hone [1] proved the integrability of the DPE by constructing its Lax pair, and showed that the equation also has single and N peakon solutions of the form (1.”
Multi-peakon solutions of the Degasperis–Procesi equation (2003, methodology): “ is not self-adjoint. The DP equation can be written as a system for u(x,t) and m(x,t): m t +m xu+3mu x = 0, (2) m= u−u xx. (3) Multi-peakon solutions of the Degasperis–Procesi equation 2 As shown in [3], this is the compatibility condition for the overdetermined linear system (∂ x−∂3)ψ= zmψ, (4) ψ ”
Stability of peakons for the Degasperis‐Procesi equation (2007, background): “Numerical simulations [13, 25] suggest that the sizes and velocities of the peakons do not change as a result of collision so these patterns are expected to be stable.”
Global Existence and Blow-Up Phenomena for the Degasperis-Procesi Equation (2006, methodology): “They also showed [ 20 ] that Eq. (1.2) has bi-Hamiltonian structure and an infinite sequence of conserved quantities, and admits exact peakon solutions which are analogous to the Camassa-Holm peakons.”