On a Camassa-Holm type equation with two dependent variables

By Gregorio Falqui. 2005

190 citations recorded by the source corpus.

We consider a generalization of the Camassa–Holm (CH) equation with two dependent variables, called CH2, introduced in a paper by Liu and Zhang (Liu S-Q and Zhang Y 2005 J. Geom. Phys. 54 427–53). We briefly provide an alternative derivation of it based on the theory of Hamiltonian structures on (the dual of) a Lie algebra. The Lie algebra involved here is the same algebra as underlies the NLS hierarchy. We study the structural properties of the hierarchy defined by the CH2 equation within the bi-Hamiltonian theory of integrable PDEs, and provide its Lax representation. Then we explicitly discuss how to construct classes of solutions, both of peakon and of algebro-geometrical type. Finally we sketch the construction of a class of singular solutions, defined by setting to zero one of the two dependent variables.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/444d272ac45175d0290286a60bc16249bd6ad50a

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Citations in the corpus

A three-component generalization of Camassa–Holm equation with N-peakon solutions (2011, background): “Recently, a two-component generalization of the CH equation has been proposed [9,20] and it has bi-Hamiltonian structure and multi-kink solutions [9,20,2].”
On the Cauchy problem for the two-component Camassa–Holm system (2011, background): “Consideration herein is the following coupled two-component Camassa–Holm shallow water system [21,26], namely, { mt + umx + 2ux m + σρρx = 0, t > 0, x ∈ R, ρt + (uρ)x = 0, t > 0, x ∈ R (1.”
On the global existence and wave-breaking criteria for the two-component Camassa-Holm system (2010, background): “1) is formally integrable [19,23,31] as it can be written as a compatibility condition of two linear systems (Lax pair) with a spectral parameter ζ , that is,”
Autonomous Ovsyannikov theorem and applications to nonlocal evolution equations and systems (2016, background): “This system is integrable (see Falqui [26] and Shabat and Alonso [65]).”
THE GEOMETRY OF THE TWO-COMPONENT CAMASSA-HOLM AND DEGASPERIS-PROCESI EQUATIONS (2010, background): “In fact, it is related to the first negative flow of the AKNS hierarchy via a reciprocal transformation [3, 12].”