Space-time Periodic Media Research Articles

Propagation dynamics of degenerate monostable equations in space–time periodic media

This paper is concerned with the multidimensional pulsating waves and spreading speed for a class of reaction–advection–diffusion equations with degenerate monostable nonlinearity in space–time periodic media. Firstly, we show that the pulsating wave with the critical speed is unique up to translation, monotone and decays to zero exponentially, while the pulsating waves with noncritical speeds decay to zero nonexponentially. Then, combining the super- and sub-solution method and the asymptotic behavior of the pulsating waves, it is proved that the spreading speed of the degenerate monostable equations with compactly supported initial data is characterized by the critical speed under appropriate assumptions.

Spreading in space–time periodic media governed by a monostable equation with free boundaries, Part 2: Spreading speed

This is Part 2 of our work aimed at classifying the long-time behavior of the solution to a free boundary problem with monostable reaction term in space–time periodic media. In Part 1 (see [2]) we have established a theory on the existence and uniqueness of solutions to this free boundary problem with continuous initial functions, as well as a spreading-vanishing dichotomy. We are now able to develop the methods of Weinberger [15,16] and others [6–10] to prove the existence of asymptotic spreading speed when spreading happens, without knowing a priori the existence of the corresponding semi-wave solutions of the free boundary problem. This is a completely different approach from earlier works on the free boundary model, where the spreading speed is determined by firstly showing the existence of a corresponding semi-wave. Such a semi-wave appears difficult to obtain by the earlier approaches in the case of space–time periodic media considered in our work here.

Unusual electromagnetic modes in space-time-modulated dispersion-engineered media

We report on electromagnetic modes in space-time-modulated dispersion-engineered media. These modes exhibit unusual dispersion relation, field profile, and scattering properties. They are generated by coupled codirectional space-time harmonic pairs and occur in space-time periodic media whose constituent materials exhibit specific dispersion. Excitation of a slab of such a medium with subluminal modulation results in periodic transfer of energy between the incident frequency and a frequency shifted by a multiple of the modulation frequency, whereas superluminal modulation generates exponentially growing anharmonics. These modes may find applications in optical mixers, terahertz sources, and other optical devices.

Curved fronts of monostable reaction-advection-diffusion equations in space-time periodic media

This paper is to study traveling fronts of reaction-diffusion equations with space-time periodic advection and nonlinearity in $\mathbb{R}^N$ with $N\geq3$. We are interested in curved fronts satisfying some ``pyramidal conditions at infinity. In $\Bbb{R}^3$, we first show that there is a minimal speed $c^{*}$ such that curved fronts with speed $c$ exist if and only if $c\geq c^{*}$, and then we prove that such curved fronts are decreasing in the direction of propagation. Furthermore, we give a generalization of our results in $\mathbb{R}^N$ with $N\geq4$.

Pulsating traveling fronts in space–time periodic media

This Note deals with the existence of pulsating traveling fronts for some reaction–diffusion equation in space–time periodic media. Under some hypotheses, there exist two speeds c ∗ and c ∗ ∗ such that there exist some pulsating traveling fronts of speed c for all c ⩾ c ∗ ∗ and that there exists no such front of speed c < c ∗ . In the case of a KPP-type reaction term, we characterize this speed with the help of a family of eigenvalues associated with the equation. Lastly, we study the dependence between this minimal speed and the coefficients of the equation. To cite this article: G. Nadin, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Reaction–diffusion equations in space–time periodic media

This Note deals with reaction–diffusion in space–time periodic media. We state some conditions for the existence, uniqueness and large-time behavior of the solutions of such equations. These conditions are related to the two generalized principal eigenvalues associated with a linearized equation and we state some properties of these quantities. To cite this article: G. Nadin, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Cerenkov and transition radiation in space-time periodic media

The solution to the problem of determining the radiation emitted by a uniformly moving charged particle in a sinusoidally space-time periodic medium is obtained. The space-time periodicity can be considered as due to a strong pump wave and is expressed as a traveling-wave-type change in the dielectric constant or the plasma density. The solution covers also the limiting case of sinusoidally stratified media. The expression and spectrum of the radiated electromagnetic field are determined for different media: dielectric, isotropic and uniaxial plasma. Depending on the nature of the medium and the velocity of the particle, the radiated field is of the Cerenkov and/or transition type. The Brillouin diagram is used extensively in understanding and determining the nature, extent, and spectrum of the different modes of radiation, and a focusing effect is also studied.

Electromagnetic wave propagation and wave-vector diagram in space-time periodic media

The electromagnetic wave propagation and wave-vector diagram in generally space-time periodic dielectric, plasma, and uniaxial plasma are studied for TE and TM waves. The case of a sinusoidal periodicity is solved numerically. Special properties due to the inhomogeneity are presented.

Wave propagation and dispersion in space-time periodic media

The dispersion characteristics of electromagnetic fields are examined for waves propagating in a medium with a permittivity which is modulated periodically with respect to time and one spatial co-ordinate. By taking a guided-wave approach, which expresses the fields of arbitrary sources in terms of a superposition of modal solutions, it is shown that many properties of the individual modes may be inferred by means of wavenumber diagrams. These diagrams are easily constructed for the special case of a vanishingly small periodic modulation. By using coupled-mode consideration, this special case is extended to serve as a good approximation for finite modulation amplitudes. In addition to yielding information on the dispersion character of modal fields, it is shown that the wavenumber diagrams may be employed to extract the modal constituents of a plane wave scattered by a layer containing a space-time periodic medium. The principal far-field components of waves excited by a localised arbitrary source embedded in such a medium may also easily be obtained from the wavenumber diagrams. The guided-wave approach, together with its associated wavenumber diagrams, is thus shown to serve as a powerful tool in analysing and understanding a large class of phenomena, which includes the diffraction of light by sound and parametric effects in nonlinear media, as well as other aspects of wave interactions in material bodies.

Group velocity in space-time periodic media

The concept of the group velocity of electromagnetic fields is examined for waves that propagate through an isotropic lossless dielectric medium whose permittivity is varying periodically with respect to both space and time. It is found that the group velocity represents the velocity of energy transport averaged over the length of a periodicity cell.