In this paper we present an analytical study of a subgrid scale turbulence model of the threedimensional magnetohydrodynamic (MHD) equations, inspired by the Navier-Stokes-α (also known as the viscous Camassa-Holm equations or the Lagrangian-averaged Navier-Stokes-α model. Specifically, we show the global well-posedness and regularity of solutions of a certain MHD-α model (which is a particular...

Abstract. Let G be a Banach Lie group modeled on the Banach space, possibly infinite dimensional, E. In this paper first we introduce Euler-Lagrange equations on the Lie group G with potential and right invariant metric. Euler-Lagrange equations are natural extensions of the geodesic equations on manifolds and Lie groups. In the second part, we study the geometry of the mechanical system of a r...

We establish the inverse spectral transform for the conservative Camassa–Holm flow with decaying initial data. In particular, it is employed to prove existence of weak solutions for the corresponding Cauchy problem.

The Poisson brackets for the scattering data of the Camassa-Holm equation are computed. Consequently, the action-angle variables are expressed in terms of the scattering data. PACS: 02.30.Ik, 05.45.Yv, 45.20.Jj, 02.30.Jr

An extension of the Camassa-Holm hierarchy is constructed in this letter. The conserved quantities of the hierarchy are studied and a recurrent formula for the integrals of motion is derived. PACS: 02.30.Ik; 05.45.Yv; 45.20.Jj; 02.30.Jr.

We define a kinetic and potential energy such that the principle of stationary action from Lagrangian mechanics yields Camassa--Holm system (2CH) as governing equations. After discretizing these energies, we use same variational to derive semi-discrete equations an approximation 2CH system. The discretizaton is only available in coordinates requires inversion discrete Sturm--Liouville operator ...