Reflection K-matrices for 19-vertex models

We derive and classify all regular solutions of the boundary Yang-Baxter equation for 19-vertex models known as Zamolodchikov-Fateev or A₁⁽¹⁾ model, Izergin-Korepin or A₂⁽²⁾ model, sl(2|1) model and the osp(2|1) model. We find that there is a general solution for A₁⁽¹⁰⁾ and sl(2|1) models. In both models it is a complete K-matrix with three free parameters. For the A₂⁽²⁾ and os(2|1) models we find three general solutions, being two complete reflection K-matrices solutions and one incomplete reflection K-matrix solution with some null entries. In both models these solutions have two free parameters. Integrable spin-1 Hamiltonians with general boundary interactions are also presented. Several reduced solutions from these general solutions are presented in the appendices.     

An analysis of the Lorenz equations: Stable exact solutions

We prove the existence of stable solutions (x(t), y(t), z(t)) to the Lorenz system, t being the time variable. These solutions are valid for b = 2σ and r > 1, b, r and σ being positive parameters, and fall into two classes C_i and C_d depending on whether (x(t), z(t)) are both monotonically increasing or decreasing functions of t. The analytic structure of the solutions is also discussed.     

Vacuum spherically symmetric solutions in f(T) gravity

Spherically symmetric static vacuum solutions have been built in f(T) models of gravity theory. We apply some conditions on the metric components; then new vacuum spherically symmetric solutions are obtained. Also, by extracting metric coefficients we determine the analytical form of f(T).     

Isovector Q balls

Finite-energy solutions rotating at a constant angular velocity in isovector space are considered within theories that possess global SO(3) symmetry (isovector theories). It is shown that, for a nonlinear O(3) model (n field), such solutions exist only in the one-dimensional case. For the isovector theory proposed by S. Weinberg in order to describe low-energy properties of the π mesons, such solutions exist at some values of parameters that appear in this theory. Some properties of these solutions are studied.     

Solitons and poles in the nonlinearity parameter

The KdVN-soliton solutions are analysed in terms of the perturbation parameter λ which governs the nonlinearity. They are generated by rational Stieltjes functionsQ ^(N) (λ), each pole of which can be associated with a soliton. The asymptotic emergence of the separate solitons follows at once from the motion of the poles along the negative real λ-axis. Successive diagonal Padé approximants ofQ ^(N) (λ) are considered. They provide a class of approximate solutions with a striking semisoliton like behaviour.     

Exploring plane-symmetric solutions in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>) gravity

The modified theories of gravity, especially the f(R) gravity, have attracted much attention in the last decade. This paper is devoted to exploring plane-symmetric solutions in the context of metric f(R) gravity. We extend the work on static plane-symmetric vacuum solutions in f(R) gravity already available in the literature [1, 2]. The modified field equations are solved using the assumptions of both constant and nonconstant scalar curvature. Some well-known solutions are recovered with power-law and logarithmic forms of f(R) models.     

The general exact bijective continuous solution of Feigenbaum's functional equation

Solutionsf:?→? of Feigenbaum's functional equationf(f(x))=α^?1 f(αx), where α≠0 is a fixed real number, account for many of the fascinating properties of the behaviour of successive iterates of (one parameter families of) nonlinear maps. In connection with the phenomenon of intermittency, interesting families of exact solutions have recently been found (for α>0). These solutions can all be derived from continuous bijective solutions which are topologically equivalent to translations. In this paper, thegeneral exact continuous bijective solution is found for any α≠0, positive or negative. In particular, it is shown that, forany α≠0, there are solutions which areinequivalent to translations. And it is shown that bijective solutions equivalent to translations exist only when 0<α<1. These results considerably enlarge the stock of available exact solutions of Feigenbaum's equation.     

Quark and strange quark matter in f(R) gravity for Bianchi type I and V space-times

Behaviors of quark matter and strange quark matter which exist in the first seconds of the early Universe in f(R) gravity are studied for Bianchi I and V universes. In this respect, we obtain exact solutions of the modified Einstein field equations by using anisotropy feature of Bianchi I and V space-times. In particular, we investigate exact f(R) functions for Bianchi I as the contribution of strange quark and quark matter. Also, we have concluded that quark matter may contribute to the early acceleration of the universe since quark matter behaves like phantom-type dark energy. Furthermore, obtained f(R) solutions represents early eras of the Universe since f(R) solutions for quark matter coincide with f(R) equations for inflation. From this point, we can reach the conclusion that quarks may be source of the early dark energy of the universe or source of little inflation due to their repulsive force.     

Solution for fractional Zakharov–Kuznetsov equations by using two reliable techniques

In this paper, the approximated analytical solutions for nonlinear dispersive fractional Zakharov-Kuznetsov (FZK(a, b, c)) equations are obtained with the help of two novel techniques, called fractional natural decomposition method (FNDM) and q-homotopy analysis transform method (q-HATM). In order to validate and illustrate the efficiency of the proposed techniques, we consider two special cases FZK(2, 2, 2) and FZK(3, 3, 3) in FZK(a, b, c). The numerical simulations have been conducted in order to verify the proposed techniques are reliable and accurate. The outcomes are revealed through the plots and tables. The comparison between the obtained solutions with the exact solutions exhibits that, both the featured schemes are efficient and effective in solving nonlinear complex problems.     

The Cauchy problem for the O(N), CP(N − 1), and GC(N,p) models

We study the Cauchy problem for the O(N), $\text{C}$P(N ? 1) and G$\text{C}$(N, p) models in n + 1 dimensional space-time. We prove the existence and uniqueness of solutions for small time intervals and for any n. In space-time dimension two, the previous solutions can be extended to all times by the method of a priori estimates. In space-time dimensions three and four, our estimates yield only partial results on the global existence problem. In all cases the solutions are required only to belong to local spaces, which means that they satisfy local regularity conditions but have no restrictions on their behaviour at infinity in space.     

A New Class of Bianchi Cosmological Models in Lyra’s Geometry

The new class of cosmological model of the early Universe filled with perfect fluid in Lyra’s geometry has been considered. We obtain two classes of exact solutions of the field equations in Lyra’s geometry with a time-dependent displacement vector. The exact solutions to the corresponding field equations are obtained in quadrature form. The cosmological parameters have been discussed in detail and it is also shown that the solutions tend asymptotically to isotropic Friedmann-Robertson-Walker cosmological model. We have also discussed the well-known astrophysical phenomena, namely the Hubble parameter H(z), luminosity distance d _L and distance modulus μ(z) with redshift.     

Monopole classical solutions and the vacuum structure in lattice gauge theories

Classical solutions corresponding to monopole-antimonopole pairs are found in 3d and 4d SU(2) and U(1) lattice gauge theories. The stability of these solutions in various theories is studied.     

(2 + 1)-Dimensional Solutions in F(R) Gravity

Motivated by the well-known charged BTZ black holes, we look for (2 + 1)-dimensional solutions of F(R) gravity. At first we investigate some near horizon solutions and after that we obtain asymptotically Lifshitz black hole solutions. Finally, we discuss about rotating black holes with exponential form of F(R) theory.     

Zero-sound branches in asymmetric nuclear matter

A polarization operator constructed in the random phase approximation is used to obtain zero-sound excitations in isospin asymmetric nuclear matter (ANM). Two families of the complex solutions ω_sτ(k),τ= p,n are presented. The imaginary part of the solutions corresponds to the damping of the collective mode due to its overlapping with the particle-hole modes and the subsequent emission of a proton (ω_sp(k)) or a neutron (ω_sn(k)). The dependence of the solutions on the asymmetry parameter is studied.     

Form factors of exponential operators and exact wave function renormalization constant in the Bullough-Dodd model

We compute the form factors of exponential operators e^kgϕ(x) in the two-dimensional integrable Bullough-Dodd model (a₂⁽²⁾ affine Toda field theory). These form factors are selected among the solutions of general non-derivative scalar operators by their asymptotic cluster property. Through analytical continuation to complex values of the coupling constant these solutions permit to compute the form factors of scaling relevant primary fields in the lightest-breather sector of integrable /gf_1,2 and /gf_1,5 deformations of conformal minimal models. We also obtain the exact wave-function renormalization constant Z(g) of the model and the properly normalized form factors of the operators ϕ(x) and : ϕ²(x) :.     

Analytic description of distortionless propagation in a resonant absorbing medium with overlapping Q(j) transitions

Analytic closed form solutions are obtained for distortionless propagation of ultra-short optical pulses through a resonant medium with overlapping Q(j) transitions for j = 2. Apart from the expected 2π solutions, we also found 0π solutions in contrast to conclusions based on numerical calculations.     

Noncommutative Wormholes in f(R) Gravity with Lorentzian Distribution

In this paper, we derive some new exact solutions of static wormholes in f(R) gravity supported by the matter possesses Lorentzian density distribution of a particle-like gravitational source. We derive the wormhole’s solutions in two possible schemes for a given Lorentzian distribution: assuming an astrophysically viable F(R) function such as a power-law form and discuss several solutions corresponding to different values of the exponent (here $F =\frac{df}{dR}$ ). In the second scheme, we consider particular form of two shape functions and have reconstructed f(R) in both cases. We have discussed all the solutions with graphical point of view.     

Dark-energy cosmological models in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">G</Emphasis>) gravity

We discuss dark-energy cosmological models in f(G) gravity. For this purpose, a locally rotationally symmetric Bianchi type I cosmological model is considered. First, exact solutions with a well-known form of the f(G) model are explored. One general solution is discussed using a power-law f(G) gravity model and physical quantities are calculated. In particular, Kasner’s universe is recovered and the corresponding f(G) gravity models are reported. Second, the energy conditions for the model under consideration are discussed using graphical analysis. It is concluded that solutions with f(G) = G^5/6 support expansion of universe while those with f(G) = G^1/2 do not favor the current expansion.     

Degenerate Sklyanin algebras

New trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for sl(2;?). These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The sl(N;?)-case is discussed.     

On finite action solutions of the nonlinear σ-model

The instanton and anti-instanton solutions of the two-dimensional O(3) σ-model are special examples of harmonic maps, which have been studied extensively in the mathematical literature. We give an elementary and self-contained proof that these solutions are the only continuous maps for which the action is finite and stationary under variations, without assuming any additional boundary conditions at infinity. An element of the proof is the vanishing of the stress tensor for a finite action solution, which actually holds true for the general O(N) σ-model. For the two-dimensional O(2l + 1) σ-model we exhibit explicit finite action solutions that do not lie in any lower dimensional sphere; the existence of such solutions has been pointed out in the mathematical literature. We also present a rigorous proof, based on Derrick's scaling argument, that there are no nonconstant finite action solutions in more than two dimensions.