On Symmetry Reduction of Nonlinear Generalization of the Heat Equation

Abstract

Reductions and classes of new exact solutions are constructed for a class of Galilei-invariant heat equations.     

Reduction and Exact Solutions of the Monge-Ampere Equation

Abstract

On the basis of a subgroup structure of the Poincaré group P(1, 3) the ansatzes which reduce the Monge–Ampere equation to differential equations with fewer independent variables have been constructed. The corresponding symmetry reduction has been done. By means of the solutions of the reduced equations some classes of exact solutions of the investigated equation have been found.     

Symmetry Reduction and Exact Solutions of the Eikonal Equation

Abstract

By means of splitting subgroups of the generalized Poincaré group P(1, 4), ansatzes which reduce the eikonal equation to differential equations with fewer independent variables have been constructed. The corresponding symmetry reduction has been done. By means of the solutions of the reduced equations some classes of exact solutions of the investigated equation have been presented.     

Symmetries and invariants for the 2D-Ricci flow model

Abstract

The paper investigates some special Lie type symmetries and associated invariant quantities which appear in the case of the 2D Ricci flow equation in conformal gauge. Starting from the invariants some simple classes of solutions will be determined.     

Plane Symmetric Inhomogeneous Cosmological Models with a Perfect Fluid in General Relativity

In this paper we investigate a class of solutions of Einstein equations for the plane- symmetric perfect fluid case with shear and vanishing acceleration. If these solutions have shear, they must necessarily be non-static. We examine the integrable cases of the field equations systematically. Among the cases with shear we find three classes of solutions. PACS No.: 04.20.-q.     

Riemann Invariants and Rank-k Solutions of Hyperbolic Systems

Abstract

In this paper we employ a “direct method” to construct rank-k solutions, expressible in Riemann invariants, to hyperbolic system of first order quasilinear di!erential equations in many dimensions. The most important feature of our approach is the analysis of group invariance properties of these solutions and applying the conditional symmetry reduction technique to the initial equations. We discuss in detail the necessary and su"cient conditions for existence of these type of solutions. We demonstrate our approach through several examples of hydrodynamic type systems; new classes of solutions are obtained in a closed form.     

Conditional Invariance and Exact Solutions of a Nonlinear System

Abstract

The Lie and Q-conditional invariance of one nonlinear system of PDEs of the third-order is searched. The ansatze have been built which reduce the PDEs system to ODEs. The classes of exact solutions of the given system are obtained. The relation between the Korteweg-de Vries equation and Harry-Dym equation has been established.     

Symmetry Reduction and Exact Solutions of the Euler–Lagrange–Born–Infeld,Multidimensional Monge–Ampere and Eikonal Equations

Abstract

Using the subgroup structure of the generalized Poincaré group P (1, 4), ansatzes which reduce the Euler–Lagrange–Born–Infeld, multidimensional Monge–Ampere and eikonal equations to differential equations with fewer independent variables have been constructed. Among these ansatzes there are ones which reduce the considered equations to linear ordinary differential equations. The corresponding symmetry reduction has been done. Using the solutions of the reduced equations, some classes of exact solutions of the investigated equation have been presented.     

Symmetry Classification of the One–Dimensional Second Order Equation of a Hydrodynamic Type

Abstract

The paper contains a symmetry classification of the one–dimensional second order equation of a hydrodynamical type L(Lu)+λLu=F (u), where L ≡ ? _t+u? _x. Some classes of exact solutions of this equation are given.     

String Cosmological Solutions in Self-Creation Theory of Gravitation

We have studied the problem of cosmic strings for Bianchi-I, II, VIII and IX string cosmological models in Barber’s (Gen. Relativ. Gravit. 14:117, 1982) second self—creation theory of gravitation. We have obtained some classes of solutions by considering different functional form for metric potentials. It is also observed that due to the presence of scalar field, the power index ‘m’ of the metric coefficients has a range of values.     

Optical properties of the quasi-two-dimensional dichalcogenides 2H-TaSe and 2H-NbSe

We present a comprehensive analysis of the optical constants of the two-dimensional dichalcogenide materials 2 H - TaSe ₂ and 2 H - NbSe ₂ , in an attempt to address the physics of two-dimensional correlated systems. The title compounds were studied over several decades in frequency, from the far-infrared to the ultraviolet. Measurements with linearly polarized light have allowed us to obtain both the in-plane and out-of-plane components of the conductivity tensor. Although the electromagnetic response of dichalcogenides is strongly anisotropic, both the in-plane and out-of-plane components of the conductivity tensor share many common features, including the presence of a well-defined metallic component, as well as a “mid-infrared band”. We discuss the implications of these results in the context of the spectroscopic results of other classes of low-dimensional conductors such as the high-temperature superconducting cuprates. In particular, the analysis of the redistribution of the spectral weight as a function of temperature, as well as the behavior of the quasiparticles relaxation rate, points to significant distinctions between the charge dynamics of dichalcogenides and other classes of low dimensional conductors. Received 28 October 2002 / Received in final form 10 March 2003 Published online 23 May 2003 RID="a" ID="a"e-mail: degiorgi@solid.phys.ethz.ch     

On multivortex solutions of the weierstrass representation

The connection between the complex sine-Gordon equation on the plane associated with a Weierstrass-type system and the possibility of constructing several classes of multivortex solutions is discussed in detail. It is shown that the amplitudes of these vortex solutions represented in polar coordinates satisfy the fifth Painlevé equation. We perform the analysis using the known relations for the Painlevé equations and construct explicit formulas in terms of the Umemura polynomials, which are τ functions for rational solutions to the third Painlevé equation. New classes of multivortex solutions to the Weierstrass system are obtained through the use of this proposed procedure.     

AdS solutions in gauge supergravities and the global anomaly for the product of complex two-cycles

Cohomological methods are applied for the special set of solutions corresponding to rotating branes in arbitrary dimensions, AdS black holes (which can be embedded in ten or eleven dimensions), and gauge supergravities. A new class of solutions is proposed, the Hilbert modular varieties, which consist of the 2n-fold product of the two-spaces H ⁿ /Γ (where H ⁿ denotes the product of n upper half-planes, H ², equipped with the co-compact action of Γ⊂SL(2,ℝ) ⁿ ) and (H ⁿ )^∗/Γ (where (H ²)^∗=H ²∪{cusp of Γ} and Γ is a congruence subgroup of SL(2,ℝ) ⁿ ). The cohomology groups of the Hilbert variety, which inherit a Hodge structure (in the sense of Deligne), are analyzed, as well as bifiltered sequences, weight and Hodge filtrations, and it is argued that the torsion part of the cuspidal cohomology is involved in the global anomaly condition. Indeed, in the presence of the cuspidal part, all cohomology classes can be mapped to the boundary of the space and the cuspidal contribution can be involved in the global anomaly condition.     

Bose-Einstein condensation in a static scalar field from the standpoint of the Goursat problem

The Goursat problem, developed by the present authors in previous papers [Ukr. Fiz. Zh. (Russ. Ed.) 27, 1602 (1982); Differentsial’nye Uravneniya 20, 302 (1984); J. Math. Phys. 33, 233 (1996)], is used to study the energy spectrum of a scalar relativistic particle in a static axisymmetric external scalar field of an attractive nature. This is obviously a model. It is shown that the problem formulated in this way has no unstable solutions, i.e., solutions increasing with time, in contrast to the Cauchy problem, where such solutions appear when the square of the particle frequency (energy) vanishes (in other words, in a Bose-Einstein condensation) Zh. éksp. Teor. Fiz. 112, 1167–1175 (October 1997)     

Three-dimensional stationary cyclic symmetric Einstein–Maxwell solutions; black holes

From a general metric for stationary cyclic symmetric gravitational fields coupled to Maxwell electromagnetic fields within the (2 + 1)-dimensional gravity the uniqueness of wide families of exact solutions is established. Among them, all uniform electromagnetic solutions possessing electromagnetic fields with vanishing covariant derivatives, all fields having constant electromagnetic invariants F_μνF^μν and T_μνT^μν, the whole classes of hybrid electromagnetic solutions, and also wide classes of stationary solutions are derived for a third-order nonlinear key equation. Certain of these families can be thought of as black hole solutions. For the most general set of Einstein–Maxwell equations, reducible to three nonlinear equations for the three unknown functions, two new classes of solutions – having anti-de Sitter spinning metric limit – are derived. The relationship of various families with those reported by different authors’ solutions has been established. Among the classes of solutions with cosmological constant a relevant place is occupied by the electrostatic and magnetostatic Peldan solutions, the stationary uniform and spinning Clement classes, the constant electromagnetic invariant branches with the particular Kamata–Koikawa solution, the hybrid cyclic symmetric stationary black hole fields, and the non-less important solutions generated via SL(2,R)-transformations where the Clement spinning charged solution, the Martinez–Teitelboim–Zanelli black hole solution, and Dias–Lemos metric merit mention.     

Does the ϱ'(1600) couple to ππ? a question of analyticity

Analyticity is exploited to distinguish between classes of π⁺π^- partial wave solutions. Fixed t and fixed u dispersion relations determine the over-all phase of the amplitude and clearly select solutions with a ?'(1600) resonance of 25% elasticity.     

Soliton Asymptotics of Rear Part of Non-Localized Solutions of the Kadomtsev-Petviashvili Equation

Abstract

We construct non-localized, real global solutions of the Kadomtsev-Petviashvili-I equation which vanish for x → ?∞ and study their large time asymptotic behavior. We prove that such solutions eject (for t → ∞) a train of curved asymptotic solitons which move behind the basic wave packet.