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Singular solutions of a modified two-component Camassa-Holm equation.

The Camassa-Holm (CH) equation is a well-known integrable equation describing the velocity dynamics of shallow water waves. This equation exhibits spontaneous emergence of singular solutions (peakons) from smooth initial conditions. The CH equation has been recently extended to a two-component integrable system (CH2), which includes both velocity and density variables in the dynamics. Although possessing peakon solutions in the velocity, the CH2 equation does not admit singular solutions in the density profile. We modify the CH2 system to allow a dependence on the average density as well as the pointwise density. The modified CH2 system (MCH2) does admit peakon solutions in the velocity and average density. We analytically identify the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data. Numerical results for the MCH2 system are given and compared with the pure CH2 case. These numerics show that the modification in the MCH2 system to introduce the average density has little short-time effect on the emergent dynamical properties. However, an analytical and numerical study of pairwise peakon interactions for the MCH2 system shows a different asymptotic feature. Namely, besides the expected soliton scattering behavior seen in overtaking and head-on peakon collisions, the MCH2 system also allows the phase shift of the peakon collision to diverge in certain parameter regimes.

Geodesic flows on semidirect-product Lie groups: geometry of singular measure-valued solutions

The EPDiff equation (or the dispersionless Camassa–Holm equation in one dimension) is a well-known example of geodesic motion on the Diff group of smooth invertible maps (diffeomorphisms). Its recent two-component extension governs geodesic motion on the semidirect product DiffⓈ, where denotes the space of scalar functions. This paper generalizes the second construction to consider geodesic motion on DiffⓈ, where denotes the space of scalar functions that take values on a certain Lie algebra (e.g. =⊗(3)). Measure-valued delta-like solutions are shown to be momentum maps possessing a dual pair structure, thereby extending previous results for the EPDiff equation. The collective Hamiltonians are shown to fit into the Kaluza–Klein theory of particles in a Yang–Mills field and these formulations are shown to apply also at the continuum partial differential equation level. In the continuum description, the Kaluza–Klein approach produces the Kelvin circulation theorem.

Reduction theory for symmetry breaking with applications to nematic systems

We formulate Euler-Poincare and Lagrange-Poincare equations for systems with broken symmetry. We specialize the general theory to present explicit equations of motion for nematic systems, ranging from single nematic molecules to biaxial liquid crystals. The geometric construction applies to order parameter spaces consisting of either unsigned unit vectors (directors) or symmetric matrices (alignment tensors). On the Hamiltonian side, we provide the corresponding Poisson brackets in both Lie-Poisson and Hamilton-Poincare formulations. The explicit form of the helicity invariant for uniaxial nematics is also presented, together with a whole class of invariant quantities (Casimirs) for two-dimensional incompressible flows.

Vlasov moments, integrable systems and singular solutions

Abstract The Vlasov equation governs the evolution of the single-particle probability distribution function (PDF) for a system of particles interacting without dissipation. Its singular solutions correspond to the individual particle motions. The operation of taking the moments of the Vlasov equation is a Poisson map. The resulting Lie–Poisson Hamiltonian dynamics of the Vlasov moments is found to be integrable is several cases. For example, the dynamics for coasting beams in particle accelerators is associated by a hodograph transformation to the known integrable Benney shallow-water equation. After setting the context, the Letter focuses on geodesic Vlasov moment equations. Continuum closures of these equations at two different orders are found to be integrable systems whose singular solutions characterize the geodesic motion of the individual particles.

Hybrid Vlasov-MHD models: Hamiltonian vs. non-Hamiltonian

This paper investigates hybrid kinetic-magnetohydrodynamic (MHD) models, where a hot plasma (governed by a kinetic theory) interacts with a fluid bulk (governed by MHD). Different nonlinear coupling schemes are reviewed, including the pressure-coupling scheme (PCS) used in modern hybrid simulations. This latter scheme suffers from being non-Hamiltonian and is unable to exactly conserve total energy. Upon adopting the Vlasov description for the hot component, the non-Hamiltonian PCS and a Hamiltonian variant are compared. Special emphasis is given to the linear stability of Alfven waves, for which it is shown that a spurious instability appears at high frequency in the non-Hamiltonian version. This instability is removed in the Hamiltonian version.

Hamiltonian approach to hybrid plasma models

The Hamiltonian structures of several hybrid kinetic-fluid models are identified explicitly, upon considering collisionless Vlasov dynamics for the hot particles interacting with a bulk fluid. After presenting different pressure-coupling schemes for an ordinary fluid interacting with a hot gas, the paper extends the treatment to account for a fluid plasma interacting with an energetic ion species. Both current-coupling and pressure-coupling MHD schemes are treated extensively. In particular, pressure-coupling schemes are shown to require a transport-like term in the Vlasov kinetic equation, in order for the Hamiltonian structure to be preserved. The last part of the paper is devoted to studying the more general case of an energetic ion species interacting with a neutralizing electron background (hybrid Hall-MHD). Circulation laws and Casimir functionals are presented explicitly in each case.

Euler-Poincare Formulation Of Hybrid Plasma Models

Three different hybrid Vlasov-fluid systems are derived by applying reduction by symmetry to Hamiltons variational principle. In particular, the discussion focuses on the Euler-Poincare formulation of three major hybrid MHD models, which are compared in the same framework. These are the current-coupling scheme and two different variants of the pressure-coupling scheme. The Kelvin-Noether theorem is presented explicitly for each scheme, together with the Poincare invariants for its hot particle trajectories. Extensions of Ertels relation for the potential vorticity and for its gradient are also found in each case, as well as new expressions of cross helicity invariants.

Formulation of the relativistic moment implicit particle-in-cell method

A new formulation is presented for the implicit moment method applied to the time-dependent relativistic Vlasov-Maxwell system. The new approach is based on a specific formulation of the implicit moment method that allows us to retain the same formalism that is valid in the classical case despite the formidable complication introduced by the nonlinear nature of the relativistic equations of motion. To demonstrate the validity of the new formulation, an implicit finite difference algorithm is developed to solve the Maxwell’s equations and equations of motion. A number of benchmark problems are run: two stream instability, ion acoustic wave damping, Weibel instability, and Poynting flux acceleration. The numerical results are all in agreement with analytical solutions.

Geometry of Vlasov kinetic moments: A bosonic Fock space for the symmetric Schouten bracket

Abstract The dynamics of Vlasov kinetic moments is shown to be Lie–Poisson on the dual Lie algebra of symmetric contravariant tensor fields. The corresponding Lie bracket is identified with the symmetric Schouten bracket and the moment Lie algebra is related with a bundle of bosonic Fock spaces, where creation and annihilation operators are used to construct the cold plasma closure. Kinetic moments are also shown to define a momentum map, which is infinitesimally equivariant. This momentum map is the dual of a Lie algebra homomorphism, defined through the Schouten bracket. Finally the moment Lie–Poisson bracket is extended to anisotropic interactions.

Vlasov moment flows and geodesics on the Jacobi group

By using the moment algebra of the Vlasov kinetic equation, we characterize the integrable Bloch-Iserles system on symmetric matrices [Bloch, A. M., Brinzănescu, V., Iserles, A., Marsden, J. E., and Ratiu, T. S., “A class of integrable flows on the space of symmetric matrices,” Commun. Math. Phys. 290, 399–435 (2009)]10.1007/s00220-009-0849-6 as a geodesic flow on the Jacobi group Jac (R2n)= Sp (R2n)ⓈH(R2n). We analyze the corresponding Lie-Poisson structure by presenting a momentum map, which both untangles the bracket structure and produces particle-type solutions that are inherited from the Vlasov-like interpretation. Moreover, we show how the Vlasov moments associated to Bloch-Iserles dynamics correspond to particular subgroup inclusions into a group central extension (first discovered by Ismagilov, Losik, and Michor [“A 2-cocycle on a group of symplectomorphisms,” Mosc. Math. J. 6, 307–315 (2006)]), which in turn underlies Vlasov kinetic theory. In the most general case of Bloch-Iserles dynamics, a g...