Nonlinear regimes in spin dynamics of superfluid 3He

We study an algebra of Poisson brackets of the Hamiltonian system defined by the nonlinear Leggett equations of spin dynamics in the A- and the B-phases of superfluid ³He. For the A-phase the Poisson algebra results in a special case of the equations of motion of a rigid body in ideal fluid; for the B-phase, in the absence of magnetic field, it allows for a reduction to a smaller Poisson algebra that provides exact solutions for the Leggett equations.     

Spectral Action for Scalar Perturbations of Dirac Operators

We investigate the leading terms of the spectral action for odd-dimensional Riemannian spin manifolds with the Dirac operator perturbed by a scalar function. We calculate first two Gilkey–de Witt coefficients and make explicit calculations for the case of n-spheres with a completely symmetric Dirac. In the special case of dimension 3, when such perturbation corresponds to the completely antisymmetric torsion, we carry out the noncommutative calculation following Chamseddine and Connes (J Geom Phys 57:121, 2006) and study the case of SU _q (2).     

Travelling Breathers with Exponentially Small Tails in a Chain of Nonlinear Oscillators

We study the existence of travelling breathers in Klein-Gordon chains, which consist of one-dimensional networks of nonlinear oscillators in an anharmonic on-site potential, linearly coupled to their nearest neighbors. Travelling breathers are spatially localized solutions which appear time periodic in a referential in translation at constant velocity. Approximate solutions of this type have been constructed in the form of modulated plane waves, whose envelopes satisfy the nonlinear Schrödinger equation (M. Remoissenet, Phys. Rev. B 33, n.4, 2386 (1986), J. Giannoulis and A. Mielke, Nonlinearity 17, p. 551–565 (2004)). In the case of travelling waves (where the phase velocity of the plane wave equals the group velocity of the wave packet), the existence of nearby exact solutions has been proved by Iooss and Kirchgässner, who have obtained exact solitary wave solutions superposed on an exponentially small oscillatory tail (G. Iooss, K. Kirchgässner, Commun. Math. Phys. 211, 439–464 (2000)). However, a rigorous existence result has been lacking in the more general case when phase and group velocities are different. This situation is examined in the present paper, in a case when the breather period and the inverse of its velocity are commensurate. We show that the center manifold reduction method introduced by Iooss and Kirchgässner is still applicable when the problem is formulated in an appropriate way. This allows us to reduce the problem locally to a finite dimensional reversible system of ordinary differential equations, whose principal part admits homoclinic solutions to quasi-periodic orbits under general conditions on the potential. For an even potential, using the additional symmetry of the system, we obtain homoclinic orbits to small periodic ones for the full reduced system. For the oscillator chain, these orbits correspond to exact small amplitude travelling breather solutions superposed on an exponentially small oscillatory tail. Their principal part (excluding the tail) coincides at leading order with the nonlinear Schrödinger approximation.     

Analytical solutions for the spin-1 Bose-Einstein condensate in a harmonic trap

The homotopy analysis method and Galerkin spectral method are applied to find the analytical solutions for the Gross-Pitaevskii equations, a set of nonlinear Schrödinger equation used in simulation of spin-1 Bose-Einstein condensates trapped in a harmonic potential. We investigate the one-dimensional case and get the approximate analytical solutions successfully. Comparisons between the analytical solutions and the numerical solutions have been made. The results indicate that they are in agreement well with each other when the atomic interaction is weakly. We also find a class of exact solutions for the stationary states of the spin-1 system with harmonic potential for a special case.     

Algebraic and Geometric Properties of Matrix Solutions of Nonlinear Wave Equations

We improve the construction of exact matrix solutions for nonlinear wave equations by using unitary anti-Hermitian and anticommuting matrices. We prove the theorem that constructs the matrix functions u _n satisfying the nonlinear wave equation for a set of special potentials. In this case, the graph of complex solution u ₁ has a soliton-like form with a finite number of coils. Exponential representation of matrix solutions u _n is associated with continuous rotations that can be used for describing intrinsic rotations and state changes of elementary particles. We also prove the theorem on the decomposition of continuous rotation (described by solution u ₂) onto three simultaneous rotations about coordinate vectors. Each of the three constructed matrix solutions u ₃ is also decomposed into the triplet of elementary matrix solutions.     

Symbolic Computation and New Soliton-Like Solutions of the 1+2D Calogero–Bogoyavlenskii–Schif Equation

In this paper many soliton-like solutions with arbitrary functions of y and t are obtained for the 1+2D Calogero-Bogoyavlenskii-Schif equation by means of some generalized ansatz and symbolic computation. Some new exact solutions are also presented from the obtained soliton-like solutions. Shock wave solution is only a special case.     

Some solutions of the 3D Laplace equation in a layer with oscillating boundary describing an array of nanotubes with applications to cold field emission. II. Irregular arrays

As in the first part (J. Brüning, S.Yu. Dobrokhotov, D.S. Minenkov, 2011), we construct a family of special solutions of the Dirichlet problem for the Laplace equation in a domain with fast changing boundary. Using these solutions, we construct an analytic model of cold field electron emission from surfaces simulating arrays of vertically aligned nanotubes. Explicit analytic formulas lead to fast computations and also allow us to study the case of random arrays of tubes with stochastic distribution of parameters. We present these results and compare them with numerical approximations given in [1].     

Evolution of Bouncing Cosmological Models Due to the Self-Gravitational Corrections

A four-dimensional timelike brane with non-zero energy density is considered as the boundary of a five dimensional Schwarzschild anti de Sitter bulk background. The self-gravitational corrections to the first Friedmann equation act as a source of stiff matter contrary to standard FRW cosmology where the charge of the black hole plays this role. In a previous related paper (Setare in Eur. Phys. J. C 47:851, [2006]), bouncing cosmology was studied, from a holographic perspective, for the very special case of a brane that is void of any intrinsic matter sources. In this paper we extend the results of (Setare in Eur. Phys. J. C 47:851, [2006]). We consider the physically relevant case in which a perfect fluid with equation of state of radiation is present on the brane. Then, we describe solutions of the braneworld theory under investigation and also determine their stability. Specifically, if we do not consider the self-gravitational corrections, the AdS black hole with zero ADM mass, and open horizon is an attractor, while, if we consider, the AdS black hole with zero ADM mass and flat horizon, and D3-brane with non-zero energy density is a repeller.     

Classical and Nonclassical Symmetries of a Generalized Boussinesq Equation

Abstract

We apply the Lie-group formalism and the nonclassical method due to Bluman and Cole to deduce symmetries of the generalized Boussinesq equation, which has the classical Boussinesq equation as an special case. We study the class of functions f(u) for which this equation admit either the classical or the nonclassical method. The reductions obtained are derived. Some new exact solutions can be derived.     

Models of Self-Financing Hedging Strategies in Illiquid Markets: Symmetry Reductions and Exact Solutions

We study the general model of self-financing trading strategies in illiquid markets introduced by Schönbucher and Wilmott (SIAM J Appl Math 61(1):232?C272, 2000). A hedging strategy in the framework of this model satisfies a nonlinear partial differential equation (PDE) which contains some function g(??). This function is deeply connected to a marginal utility function. We describe the Lie symmetry algebra of this PDE and provide a complete set of reductions of the PDE to ordinary differential equations (ODEs). In addition, we show the way how to describe all types of functions g(??) for which the PDE admits an extended Lie group. Two of these special type functions correspond to the models introduced before by different authors, whereas one is new. We clarify the connection between these three special models and the general model for trading strategies in the illiquid markets. We also apply the Lie group analysis to the new special case of the PDE describing the self-financing strategies. For the general model, as well as for the new special model, we provide the optimal systems of subalgebras and study the complete set of reductions of the PDEs to ODEs. We provide explicit solutions to the new special model in all reduced cases. Moreover, in one of the cases the solutions describe power derivative products.     

(G′/G)展开法在高维非线性物理方程中的新应用

将（G′/G)展开首次法扩展到构造高维非线性物理方程的精确非行波通解、研究解的特殊孤子结构和混沌行为. 作为（G′/G)展开法的新应用, 获到了（3+1)维非线性Burgers系统的新非行波通解, 对通解中的任意函数进行适当的设置, 探讨了特殊孤子结构的激发和演化、解的混沌行为和演化. 关键词： G′/G)展开法')" href="#">（G′/G)展开法 Burgers系统 孤子结构 混沌行为     

Invariant resolutions for several Fueter operators

A proper generalization of complex function theory to higher dimension is Clifford analysis and an analogue of holomorphic functions of several complex variables were recently described as the space of solutions of several Dirac equations. The four-dimensional case has special features and is closely connected to functions of quaternionic variables. In this paper we present an approach to the Dolbeault sequence for several quaternionic variables based on symmetries and representation theory. In particular we prove that the resolution of the Cauchy–Fueter system obtained algebraically, via Gröbner bases techniques, is equivalent to the one obtained by R.J. Baston (J. Geom. Phys. 1992).     

All free of complex expansion null orbits type-D electrovac solutions with λ

The free of complex expansion type-D solutions of Einstein-Maxwell equations with cosmological constant possessing a noninvertible group of local isometries with null orbits for the alignment of the general electromagnetic field along the doubleD-P directions are presented. These solutions are endowed with five continuous parameters, and are found to be a special case of the Carter non-null orbits metricB(–).     

Binary ionic mixtures and a solvable model

The theory of solutions of McMillan and Mayer is applied to the jellium model of a binary ionic mixture: two species of charged particles, with charges e and Ze, immersed in a neutralizing background. The density ρ₂ of the particles of charge Ze is considered as small, and is used as an expansion parameter. The free energy, the pair distribution functions, the internal energy, and the pressure of the mixture are expressed as power series in ρ₂; the coefficients are integrals of correlation functions defined in the system at ρ₂=0 (the reference system). Explicit expressions are obtained in the two-dimensional case, at a special temperature, since in that case the reference system (the two-dimensional, one-component plasma) is a solvable model.     

Some remarks on the Hijazi inequality and generalizations of the Killing equation for spinors

We generalize the well-known lower estimates for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold proved by Friedrich [Math. Nachr. 97 (1980) 117–146] and Hijazi [Math. Phys. 104 (1986) 151–162; J. Geom. Phys. 16 (1995) 27–38]. The special solutions of the Einstein–Dirac equation constructed recently by Friedrich/Kim are examples for the limiting case of these inequalities. The discussion of the limiting case of these estimates yields two new field equations generalizing the Killing equation as well as the weak Killing equation for spinor fields. Finally, we discuss the two-and three-dimensional case in more detail.     

(G′/G)展开法和(2+1)维非对称Nizhnik-Novikov-Veselov系统的新精确解

通过引入并扩展 （G′/G）展开法, 构造出（2+1）维非对称Nizhnik-Novikov-Veselov系统的三种形式的新精确通解: 双曲函数通解, 三角函数通解, 有理函数通解. 当双曲函数通解中的常数取特定值时, 通解变为相应孤立波解. 关键词： G′/G）展开法')" href="#">（G′/G）展开法 （2+1）维非对称Nizhnik-Novikov-Veselov系统 精确解 孤立波解     

Thermal entanglement in the anisotropic Heisenberg model with Dzyaloshinskii-Moriya interaction in an inhomogeneous magnetic field

The thermal entanglement of a two-qubit anisotropic Heisenberg XXZ chain with Dzyaloshinskii-Moriya (DM) interaction in an inhomogeneous magnetic field was studied in detail. The effects of the DM parameter, external magnetic field (B), a parameter b which controls the inhomogeneity of B and the bilinear interaction parameters J_x = J_y ≠ J_z (the anisotropic case) on the concurrence (C) was formulated and studied in detail. The behaviors of the concurrences for the cases between (J = J_z = 1) and (J = -1,J_z = 1) and, (J = J_z = -1) and (J = 1,J_z = -1) at the ground state and at the thermal equilibrium are exactly the same. It was found that for the antiferromagnetic (AFM) case the entanglements persist to higher temperatures in comparison with the ferromagnetic (FM) case. In addition, the AFM case presents a special point at which the nonzero concurrences are all equal at some special temperatures. The further properties will be given in the text.     

Flat Steady States in Stellar Dynamics – Existence and Stability

We consider a special case of the three dimensional Vlasov–Poisson system where the particles are restricted to a plane, a situation that is used in astrophysics to model extremely flattened galaxies. We prove the existence of steady states of this system. They are obtained as minimizers of an energy-Casimir functional from which fact a certain dynamical stability property is deduced. From a mathematics point of view these steady states provide examples of singular solutions of the three dimensional Vlasov–Poisson system. Received: 26 October 1998 / Accepted: 23 February 1999     

Accelerating black holes in anti-de Sitter universe

A physical interpretation of theC-metric with a negative cosmological constantΛ is suggested. Using a convenient coordinate system it is demonstrated that this class of exact solutions of Einstein’s equations describes uniformly accelerating (possibly charged) black holes in anti-de Sitter universe. Main differences from the analogous de Sitter case are emphasised. Dedicated to my academic teacher Prof. J. Bičák on the occasion of his 60th birthday. This work was supported by the grant GACR-202/99/0261 of the Czech Republic and GAUK 141/2000 of Charles University in Prague.     

Chaos and dynamics of spinning particles in Kerr spacetime

We study chaos dynamics of spinning particles in Kerr spacetime of rotating black holes use the Papapetrou equations by numerical integration. Because of spin, this system exists many chaos solutions, and exhibits some exceptional dynamic character. We investigate the relations between the orbits chaos and the spin magnitude S, pericenter, polar angle and Kerr rotation parameter a by means of a kind of brand new Fast Lyapulov Indicator (FLI) which is defined in general relativity. The classical definition of Lyapulov exponent (LE) perhaps fails in curve spacetime. And we emphasize that the Poincaré sections cannot be used to detect chaos for this case. Via calculations, some new interesting conclusions are found: though chaos is easier to emerge with bigger S, but not always depends on S monotonically; the Kerr parameter a has a contrary action on the chaos occurrence. Furthermore, the spin of particles can destroy the symmetry of the orbits about the equatorial plane. And for some special initial conditions, the orbits have equilibrium points.