Poisson schemes for Hamiltonian systems on Poisson manifolds

Symmetries of Hamiltonian systems on symplectic and Poisson manifolds

Poisson Algebras and 3D Superintegrable Hamiltonian Systems

Using a Poisson bracket representation, in 3D, of the Lie algebra sl(2), we first use highest weight representations to embed this into larger Lie algebras. These are then interpreted as symmetry and conformal symmetry algebras of the “kinetic energy”, related to the quadratic Casimir function. We then consider the potentials which can be added, whilst remaining integrable, leading to families ...

From Hamiltonian Systems to Poisson Geometry

We introduce Hamiltonian systems and derive an important stability result, along with giving some physical motivation. We then move onto the generalization of these systems found in symplectic geometry. Next we consider symplectic geometry’s natural generalization, Poisson geometry. After giving some definitions we present the motivating example of the torqueless Euler equations. These motivate...

On Runge-Kutta discontinuous Galerkin schemes for Vlasov-Poisson systems

In this paper we consider Runge-Kutta discontinuous Galerkin (RKDG) schemes for Vlasov-Poisson systems that model collisionless plasmas. One-dimensional systems are emphasized. The RKDG method, originally devised to solve conservation laws, is seen to have excellent conservation properties, be readily designed for arbitrary order of accuracy, and capable of being used with a positivity-preservi...

On a class of nonlinear fractional Schrödinger-Poisson systems

In this paper, we are concerned with the following fractional Schrödinger-Poisson system:    (−∆s)u + V (x)u + φu = m(x)|u|q−2|u|+ f(x,u), x ∈ Ω, (−∆t)φ = u2, x ∈ Ω, u = φ = 0, x ∈ ∂Ω, where s,t ∈ (0,1], 2t + 4s > 3, 1 < q < 2 and Ω is a bounded smooth domain of R3, and f(x,u) is linearly bounded in u at infinity. Under some assumptions on m, V and f we obtain the existence of non-trivial so...