In this paper we examine the evolution of solutions, of a recently-derived system of cross-coupled Camassa-Holm equations, that initially have compact support. The analytical methods which we employ provide a full picture for the persistence of compact support for the momenta. For the solutions of the system itself, the answer is more convoluted, and we determine when the compactness of the support is lost, replaced instead by an exponential decay rate.

The Kepler problem for planetary motion is a two-body dynamic model with an attractive force obeying the inverse square law, and has a direct analogue in any dimension. While the magnetized Kepler problems were discovered in the late 1960s, it is not clear until recently that their higher dimensional analogues can exist at all. Here we present a possible route leading to the discovery of these high dimensional magnetized models.

A mapping between the stationary solutions of nonlinear Schrödinger equations with real and complex potentials is constructed and a set of exact solutions with real energies are obtained for a large class of complex potentials. As specific examples we consider the case of dissipative periodic soliton solutions of the nonlinear Schrödinger equation with complex potential.

The analysis of some exact solutions of Einstein equations, describing gravitational waves produced by light, shows that there can be repulsion between light beams. This is due only to the spin-1 character of the solutions and not to the introduction of other inputs external to General Relativity. Cosmological relativistic jets give an example of a such phenomenon.

In this work we consider conserved properties of the vector Nonlinear Schrödinger Equations for linearly polarized solitons in the initial configuration. We derive analytic formulae for the mass, pseudomomentum and energy and compare results with the discrete formulae based on a conservative fully implicit finite-difference scheme in complex arithmetic.

Relativistic hyperbolic geometry is a model of the hyperbolic geometry of Lobachevsky and Bolyai in which Einstein addition of relativistically admissible velocities plays the role of vector addition. The adaptation of barycentric coordinates for use in relativistic hyperbolic geometry results in the relativistic barycentric coordinates. The latter are covariant with respect to the Lorentz transformation group just as the former are covariant with respect to the Galilei transformation group. Furthermore, the latter give rise to hyperbolically convex sets just as the former give rise to convex sets in Euclidean geometry. Convexity considerations are important in non-relativistic quantum mechanics where mixed states are positive barycentric combinations of pure states and where barycentric coordinates are interpreted as probabilities. In order to set the stage for its application in the geometry of relativistic quantum states, the notion of the relativistic barycentric coordinates that relativistic hyperbolic geometry admits is studied.

In this paper, we consider the rigid spacecraft with an internal rotor as a regular point of reducible regular controlled Hamiltonian (RCH) system. In the cases of coincident and non-coincident centers of buoyancy and gravity, we give explicitly the equations of motion and Hamilton-Jacobi equations of reduced spacecraft-rotor system on the symplectic leaves by calculation in detail, which show the effect on controls in regular symplectic reduction and Hamilton-Jacobi theory respectively.