Two-component generalizations of the Camassa–Holm equation

Abstract

Recommended by Tamara Grava AbstractA classification of integrable two-component systems of non-evolutionary partial differential equations that are analogous to the Camassa-Holm equation is carried out via the perturbative symmetry approach. Independently, a classification of compatible pairs of Hamiltonian operators of specific forms is carried out, in order to obtain bi-Hamiltonian structures for the same systems of equations. Using reciprocal transformations, some exact solutions and Lax pairs are also constructed for the systems considered.

Remarks on certain two-component systems with peakon solutions

“…which is one of the coupled cubic integrable systems derived recently in [19]. After rescaling the dependent variables, this corresponds to the system (7) obtained in [31], via the Miura map (12); the equation (6) is a reduction of this system to a scalar equation.…”

“…This is a situation where the equation (28) trivially decouples from (26), with κ T = 0 implying that κ is an arbitrary function of X. For some exact solutions of the system (29), see [19]. Another exceptional situation arises by taking a reduction to a scalar equation with H = ku, v = ℓ, for k, ℓ constant, so that B = n = ℓ.…”

“…The geometrical interpretation of (1) as an Euler-Poincaré equation naturally generalizes to diffeomorphisms in two or more dimensions, and the analogues of the weak solutions (2) can be applied to the problem of template matching in computational anatomy [15]; but in general such higher-dimensional extensions do not to preserve integrability. Further research on the one-dimensional case has been concerned with the derivation [12,27] and classification [26] of integrable scalar equations analogous to (1), as well as the search for suitable two-component or multi-component analogues [6,10,16,19,32,31,33,34]. From the analytical point of view, there is also considerable interest in finding dispersive equations with higher order nonlinearity, which (despite not being integrable) display similar features in the form of peakons, wave breaking and one or more higher conservation laws [1,14].…”

Liouville Correspondences between Integrable Hierarchies

“…which arises as an integrable shallow water model [6]. The system (1.5) is of significant interest, since it exhibits nonlinear interactions between the free surface and the horizontal velocity components, and can model the phenomenon of wave breaking; see, for example, [12,13,14,19,20,23,24,47,56,58,59]. The 2CH system (1.5) is completely integrable and arises from the compatibility condition of the Lax-pair formulation [6]…”

Local-in-space blow-up criteria for two-component nonlinear dispersive wave system

“…It is worth to note that the papers [13] and [26] incorporate vorticity into the fluid model (physically, vorticity is vital for incorporating the ubiquitous effects of currents and wave-current interactions in fluid motion, also the mathematical analysis of the full-governing equations for water waves with vorticity is of particular interest). In the paper [35] a classification of integrable two-component systems of non-evolutionary partial differential equations that are analogous to the Camassa-Holm equation is carried…”