Vortex dynamics in R⁴

Banavara N. Shashikanth

The author studies the vortex dynamics of Euler's equations for a constant density fluid flow in R⁴ with special focus on singular Dirac delta distributions of the vorticity supported on two-dimensional surfaces (membranes). The self-induced velocity field of a membrane has a logarithmic divergence. A regularization done via the LIA then shows that the regularized velocity field is proportional to the mean curvature vector field of the membrane rotated by 90° in the plane of normals. The author also presents a Hamiltonian structure for the regularized self-induced motion of the membrane. The dynamics of the four-form ω ∧ ω is examined and it is shown that Ertel's vorticity theorem in R³ , for the constant density case, is a special case of this dynamics.

J. Math. Phys. 53, 013103 (2012)

Composite parameterization and Haar measure for all unitary and special unitary groups

Christoph Spengler, Marcus Huber, and Beatrix C. Hiesmayr

The authors adopt the concept of the composite parameterization of the unitary group U(d) to obtain a novel parameterization of the special unitary group SU(d), and furthermore, consider the Haar measure in terms of the introduced parameters. The authors show that the well-defined structure of the parameterization leads to a concise formula for the normalized Haar measure on U(d) and SU(d).

J. Math. Phys. 53, 013501 (2012)

The stochastisation hypothesis and the spacing of planetary systems

Jacky Cresson

The stochastisation hypothesis aims to provide a framework to deal with physical systems in random environment.It is applied here in two different cases: in the study of the dynamics of a protoplanetary nebula and in the chaotic long-term behaviour of a generic planetary system using previous works of Albeverio et al.

J. Math. Phys. 52, 113502 (2011)

On a generalization of Jacobi's elliptic functions and the double sine-Gordon kink chain

Michael Pawellek

The special properties of these functions are discussed, addition theorems are presented, and a list of indefinite integrals are given. As a physical application, it is shown that periodic kink solutions (kink chains) of the double sine-Gordon model can be described in a canonical form in terms of generalized Jacobi functions.

J. Math. Phys. 52, 113701 (2011)

Fractal structure of ferromagnets: The singularity structure analysis

Victor K. Kuetche, Thomas B. Bouetou, and Timoleon C. Kofane

Following the Weiss-Tabor-Carnevale approach designed for studying the integrability properties of nonlinear partial differential equations, the singularity structure of a (2+1)-dimensional wave-equation describing the propagation of polariton solitary waves in a ferromagnetic slab were investigated.

J. Math. Phys. 52, 092903 (2011)