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Center for Nonlinear Studies |
Since the advent of the theory of electromagnetic fields, more than a century ago, waves have been described as a flow of energy, governed by suitable transport equations in vector form, namely Maxwell's equations. In void, the electric and magnetic fields (E and B, respectively) are transversally oriented with respect to the direction of propagation, and their envelope produces a sequence of wave-fronts. This is in agreement with the fact that the energy develops according to the direction of the vector product ExB, otherwise known as Poynting vector. On the other hand, the dynamical behavior of a compressible non-viscous fluid is well described by Euler's equation, where, in principle, the velocity vector field (denoted by V) might not be necessarily related to a real material fluid. In particular, one could replace the mass density by a sort of charge density. Therefore, the temptation to describe electromagnetic and velocity fields, through the coupling of the respective modelling equations, is well motivated. We present a system of equations in three independent vector unknowns: E, B, V. In pure void, the electric and magnetic fields follow both Faraday's law and Ampere's law, where a current, flowing at velocity V, is supposed to be naturally associated with the movement of the wave-fronts. In order to close the system, V solves Euler's equation with an added forcing term E+VxB, perfectly analogous to that expressing Lorentz's law. In this way, the three entities E, B, V turn out to be strictly entangled. Such a model allows for a very large space of solutions. Moreover, it displays numerous conservation and invariance properties, all deducible from a standard analysis. An interesting subspace of solutions is the one where the third equation is reduced to E+VxB=0, which means that no acceleration is acting on the wave, so that the flow is somehow laminar. For this circumstance, the solutions (called free-waves), perfectly follow the laws of geometrical optics, ruled by the eikonal equation. Free-waves also include solitary electromagnetic pulse with compact support almost of any shape, intensity, frequency and polarization. Such a result, never achieved before, reopens the path to a serious discussion on photons, the duality wave-particle and the quantum properties of matter. Other interesting solutions (not of free-wave type) are explicitly available. Since our electromagnetic radiations actually behave as a fluid, they can be constrained to evolve in bounded regions of space, such as vortex rings (or consider for instance the so called whispering galleries). Thus, compatibly with the new model equations, an electromagnetic pulse can travel along a straight path, interact with a suitable obstacle and, if the energy allows for it, form stable circular orbits. This stored energy can be released under suitable circumstances showing a kind of quantization depending on the original frequency of rotation.
Host: Gianmarco Manzini (gmanzini@lanl.gov)