We show that the porous medium equation has a gradient ow structure which is both physically and mathematically natural. In order to convince the reader that it is mathematically natural, we show the time asymptotic behavior can be easily understood in this framework. We use the intuition and the calculus of Riemannian geometry to quantify this asymptotic behavior.

We present a systematic and unifying treatment of the problem of spontaneous nucleation of particle-antiparticle pairs in a (2+1)-dimensional system due to a static and uniform electromagnetic-like field, in the presence of quantum dissipation. We first describe a direct derivation of the Caldeira-Leggett type of mechanism for quantum dissipation within the context of string theory and of the e...

The exact semiclassical wave function for a tunneling electron, coupled to the degrees of freedom of the host medium, is constructed. In our derivation the tunneling path is divided into an infinite set of infinitezimal segments and the decaying solutions of the stationary Shrödinger equations are matched for each neighboring segment from the beginning to the end of the tunneling path. This sol...

In this paper, we carefully re-examine the chaotic RF model, first studied by Rabinovich & Fabrikant [1979], and found many new and rich complex dynamics of the model that were mostly not reported before. The chaotic RF model has proved to be a great challenge to classical numerical methods, in the sense that most classical numerical methods have not been very successful in studying the complex...

We demonstrate the existence of stable toroidal dissipative solitons with the inner phase field in the form of rotating spirals, corresponding to vorticity S=0, 1, and 2, in the complex Ginzburg-Landau equation with the cubic-quintic nonlinearity. The stable solitons easily self-trap from pulses with embedded vorticity. The stability is corroborated by accurate computation of growth rates for p...

The Bruggeman formalism provides an estimate of the effective permittivity of a particulate composite medium comprising two component mediums. The Bruggeman estimate is required to lie within the Wiener bounds and the Hashin–Shtrikman bounds. Considering the homogenization of weakly dissipative component mediums characterized by relative permittivities with real parts of opposite signs, we show...

The dispersion relations for conventional uniaxial dielectric mediums may be characterized as elliptical or elliptical–like, according to whether the medium is nondissipative or dissipative, respectively. However, under certain constitutive parameter regimes, the dispersion relations may be hyperbolic or hyperbolic–like. We investigate planewave propagation in a hyperbolic/hyperbolic–like uniax...

On total reflection from a half–space filled with an isotropic, homogeneous, weakly dissipative, dielectric–magnetic medium with negative phase–velocity (NPV) characteristics , it is shown here that a linearly polarized beam can experience either a negative or a positive Goos–Hänchen shift. The sign of the shift depends on the polarization state of the beam as well as on the signs of the real p...