Nonlinear evolution equations associated with the chiral field spectral problem

Abu Mohamed Alhasan

We establish a statistical-measures approach to describe the spectral-spatial analysis of the dynamical atom-field couplings associated with two-pair of pulses propagation in multilevel atomic media. The statistical measures are functional measures that depend on the collective coupling as well as on the eigenvalues of the Nonlinear-Liouville equation. The Nonlinear-Liouville equation is a nonlinear evolution equation which is a second-order differential equation in time. It describes the multi-wave mixing process within the atomic system, as well as coherent oscillations. The proposed spectral estimators prevail spatial multi-peak structure, which depends on the reduction or the enhancement of an effective Rabi-frequency. Decomposition of collective oscillations provides an intuitive guess to special-weight of the field's area. We introduce two functional wavelets to simulate the atomic polarization response. One of these functionals exposes the dependence on the multicomponent sine-Gordon equation for the atomic polarization, which is adequate at relatively small propagation distances. The second one is composed of combined sine and cosine functionals on the area of the fields. The cosine functional reflects the significance of the atomic inversion on the polarization response to electromagnetic field excitations. The proposed functional-wavelets inducts new sources for soliton features to the transition-radiation propagation associated with Maxwell-Bloch equations.

Arthur Koschany, Josef Kuffer, Gustav M. Obermair +1 more

Anand Aruna Kumar

We investigate the stationary amplitude and current properties of a charged particle in a uniform magnetic field within the Bohm--Madelung formulation. The analysis is organised around two complementary questions: the effect of steady currents on amplitude evolution, and the influence of the singular radial behaviour on the energy spectrum of the Landau problem. For the zero-current system, corresponding to the standard Landau problem, the Hamilton--Jacobi equations lead to an Ermakov--Pinney reference structure for the separated amplitudes. The azimuthal sector additionally admits periodic branches in which the magnetic flux is constrained by the angular periodicity condition. Allowing nonvanishing component currents under global current conservation introduces nonlinear amplitude evolution, most prominently through the magnetic-vector-potential coupling in the azimuthal sector. Under the minimum choice of equal and opposite radial and azimuthal currents, the radial contribution remains analytically tractable, while the nonlinear angular momentum and action are bounded through two complementary solvable comparison equations. The Bohmian formulation also makes it possible to examine directly the singular structure of the radial energy equation. A Fisher-based regularisation associated with the current-branching construction is shown to closely parallel the role of Langer- and JWKB-type radial regularisation, while arising from the Bohm--Madelung amplitude--momentum description itself. The resulting modification of the radial index re-orders the Landau energy spectrum and lifts the corresponding azimuthal degeneracy. Keywords: Landau problem; Bohm--Madelung mechanics; nonlinear current dynamics; Ermakov--Pinney equation; radial regularisation; comparison bounds; spectral splitting.

F.M. Ciaglia, F. Di Cosmo, A. Figueroa +5 more

Pierre-A. Vuillermot

In this article we investigate from the point of view of spectral theory the problem of relaxation to thermodynamical equilibrium of a quantum harmonic oscillator interacting with a radiation field. Our starting point is a system of infinitely many Pauli master equations governing the time evolution of the occupation probabilities of the available quantum states. The system we consider is derived from the evolution equation for the reduced density operator obtained after the initial interaction of the oscillator with the radiation field, the latter acting as a heat bath. We provide a complete spectral analysis of the infinitesimal generator of the equations, showing thereby that it generates an infinite-dimensional system of hyperbolic type. This implies that every global solution to the equations converge exponentially rapidly toward the corresponding Gibbs equilibrium state. We also provide a complete spectral analysis of a linear pencil naturally associated with the Pauli equations. All of our considerations revolve around the notion of compactness, more specifically around the existence of a new compact embedding result involving a space of sequences of Sobolev type into a weighted space of square-integrable summable sequences.

Máximo Augusto Agüero Granados, Vladmir G. Makhankov