### Regular Series

#### Vol. 45 (2014), No. 6, pp. 1131 – 1260

Solitary Waves, Shock Waves and Singular Solitons of Gardner’s Equation for Shallow Water Dynamics

Acta Phys. Pol. B 45, 1135 (2014)

article page: 1135 (contains 11 pages) •

abstract

The paper addresses the dynamics of shallow water waves that are governed by the Gardner equation, that is a generalized version of the well-known Korteweg–de Vries equation. Exact solutions are obtained in presence of shoaling and advection terms with power law nonlinearity. The paper integrates the equation by the aid of $G^{\prime }/G$-expansion method. This approach reveals singular soliton as well as shock wave solutions to the model. The solution existence criteria, also known as constraint conditions, are also displayed.

A Vortex Model for Rotating Compact Objects

Acta Phys. Pol. B 45, 1147 (2014)

article page: 1147 (contains 10 pages) •

abstract

In this paper, a rotating stationary solution of the vacuum Einstein equations with a cosmological constant is exhibited which reduces to de Sitter’s interior cosmological solution when the angular momentum goes to zero. This solution is locally isomorphic to de Sitter space, but as one approaches the axis of rotation, it has a novel feature: a conical event horizon. This suggests that, in reality, rotating compact objects have a vortex structure similar to that conjectured for rotating superfluid droplets. In the limit of slow rotation, the vortex core would be nearly cylindrical and the space-time inside the core would be Gödel-like. The exterior space-time will resemble the Kerr solution for equatorial latitudes, but significant deviations from Kerr are expected for polar latitudes.

Spontaneous Transitions in Deterministic Networks

Acta Phys. Pol. B 45, 1157 (2014)

article page: 1157 (contains 9 pages) •

abstract

The neural assemblies undergo spontaneous changes between various dynamical states characterized usually by spiking or bursting at a single neuron level. These microscopic states contribute to a global neural dynamics that may be measured in a form of electric signal referred to as a local field potential. Here, we present a model neural network composed with nodes exhibiting autonomous spiking dynamics. We show that under a particular coupling configuration and slight mismatches between the nodes, the neural network exhibits deterministic transitions between two possible configurations of clusters. The clusters, composed of two neurons each, differ in internal (always chaotic) dynamics as well as in synchronization properties. Such clusters features may contribute to a temporal increase or decrease of local field potential in the neural network, and thus give an insight into the possible mechanisms of the spontaneous brain transitions. We consider two different models for nodes, namely, forced FitzHugh–Nagumo equations and Rulkov map, and show that the presented results are node-type independent. Finally, we propose a mechanism explaining the origin of these transitions.

The Standard Model as a Low-energy Effective Theory: What is Triggering the Higgs Mechanism?

Acta Phys. Pol. B 45, 1167 (2014)

article page: 1167 (contains 48 pages) •

abstract

The discovery of the Higgs by ATLAS and CMS at the LHC not only provided the last missing building block of the electroweak Standard Model, the mass of the Higgs has been found to have a very peculiar value, about 126 GeV, which is such that vacuum stability may be extending up to the Planck scale. We emphasize the consequences for the running masses and we reconsider the role of quadratic divergences. A change of sign of the coefficient of the quadratically divergent terms, showing up at about $\mu _0\sim 1.4 \times 10^{16}$ GeV, may be understood as a first order phase transition restoring the symmetric phase in the early universe, while its large negative values at lower scales trigger the Higgs mechanism. Running parameters evolve in such a way that the symmetry is restored two orders of magnitude below the Planck scale. As a consequence, the electroweak phase transition takes place near the scale $\mu _0$ much closer to the Planck scale than anticipated so far. The SM Higgs system and its phase transition plays a key role for the inflation of the early universe. Dark energy triggering inflation is provided by the huge bare Higgs mass term and a Higgs induced vacuum density in the symmetric phase at times before the electroweak phase transition takes place.

Higgs Inflation and the Cosmological Constant

Acta Phys. Pol. B 45, 1215 (2014)

article page: 1215 (contains 40 pages) •

abstract

We discuss the impact of the Higgs discovery and its revealing a very peculiar value for the Higgs mass. It turns out that the Higgs not only induces the masses of all SM particles, the Higgs, given its special mass value, is the natural candidate for the inflaton and, in fact, is ruling the evolution of the early universe, by providing the necessary dark energy which remains the dominant energy density. In a previous paper, I have shown that running couplings not only allow us to extrapolate SM physics up to the Planck scale, but equally important they are triggering the Higgs mechanism when the universe cools down to lower temperatures. This is possible by the fact that the bare mass term in the Higgs potential changes sign at about $\mu _0\simeq 1.4\times 10^{16}$ GeV and in the symmetric phase is enhanced by quadratic terms in the Planck mass. The huge Higgs mass term together with a Higgs induced vacuum energy play a key role in triggering inflation. We show how different terms contributing to the Higgs Lagrangian are affecting inflation. Miraculously, the huge difference between bare and renormalized cosmological constant is nullified either by the running of the SM couplings or by vacuum rearrangement somewhat before the Higgs phase transition takes place. This solves the notorious cosmological constant problem. The role of the Higgs in reheating and baryogenesis is discussed.

Ginzburg–Landau Model in Three Dimensional Lobachevsky Space

Acta Phys. Pol. B 45, 1255 (2014)

article page: 1255 (contains 6 pages) •

abstract

We consider the Ginzburg–Landau model of a superconductor in three dimensional Lobachevsky space. Generally, covariant Ginzburg–Landau equations were derived and two types of solutions of these equations were obtained: with a flat and cylindrically symmetric boarders of superconductors. The first case is considered in the quasi-Cartesian coordinates system and it is shown that when the radius of curvature of Lobachevsky space $\rho$ is less than a double quantity of the London penetration depth $\lambda$, magnetic field might increase with penetration depth. In the second case, which studies cylindrically symmetric superconductor, it is shown that magnetic field depends on two coordinates: depth penetration and coordinate along the surface of the superconductor. When radius of curvature $\rho$ of Lobachevsky space goes to infinity, derived solutions and equations will go to the usual Ginzburg–Landau model in flat space.

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