Integrable and non-integrable equations with peakons

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

146 citations recorded by the source corpus.

We consider a one-parameter family of non-evolutionary partial differential equations which includes the integrable Camassa-Holm equation and a new integrable equation first isolated by Degasperis and Procesi. A Lagrangian and Hamiltonian formulation is presented for the whole family of equations, and we discuss how this fits into a bi-Hamiltonian framework in the integrable cases. The Hamiltonian dynamics of peakons and some other special finite-dimensional reductions are also described.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/9e01be3c0d19d4ba894c1682fa03410bf7f6ad43

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In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Inverse scattering transform for the Degasperis–Procesi equation (2010, background): “The dynamics of the peakon interactions for DP was elucidated in [13, 45, 46].”
Well-posedness, wave breaking and peakons for a modified μ-Camassa-Holm equation (2014, methodology): “Wave breaking phenomena and global existence of solutions for the Degasperis–Procesi equation were investigated in [6,16,18,19,34,35].”
Integrable Evolution Equations on Spaces of Tensor Densities and Their Peakon Solutions (2009, unknown): “[DHH2].”
ON THE UNIQUENESS OF DISCONTINUOUS SOLUTIONS TO THE DEGASPERIS–PROCESI EQUATION (2007, background): “By rescaling, shifting the dependent variable, and finally applying a Galilean boost, equation (1.3) can be transformed into the form (1.1), see [ 6 , 7].”
Multi-peakon solutions of the Degasperis–Procesi equation (2003, background): “Degasperis, Holm and Hone [3, 4 ] proved that the case b = 3, which they called the Degasperis– Procesi (DP) equation, is also integrable and admits multi-peakon solutions.”
On the well-posedness of the Degasperis-Procesi equation (2006, methodology): “2 The research of K.H. Karlsen is supported by an Outstanding Young Investigators Award and by the European network HYKE, Contract HPRN-CT-2002-00282.”
A Two-component Generalization of the Camassa-Holm Equation and its Solutions (2005, background): “has been studied from various point of views in, for example, [1,3,10,14,18,19,22, 24,30–32] and references therein.”