Index of content:
Volume 57, Issue 11, November 2016
We consider the 3-body problem of celestial mechanics in Euclidean, elliptic, and hyperbolic spaces and study how the Lagrangian (equilateral) relative equilibria bifurcate when the Gaussian curvature varies. We thus prove the existence of new classes of orbits. In particular, we find some families of isosceles triangles, which occur in elliptic space.
- Partial Differential Equations
Well posedness and exponential stability in a wave equation with a strong damping and a strong delay57(2016); http://dx.doi.org/10.1063/1.4966551View Description Hide Description
In this paper, we consider a wave equation with a strong damping and a strong constant (respectively, distributed) delay. We prove the well-posedness and establish an exponential decay result under a suitable assumption on the weight of the damping and the weight of the delay.
Multiplicity and concentration of positive solutions for a class of quasilinear problems through Orlicz-Sobolev space57(2016); http://dx.doi.org/10.1063/1.4966534View Description Hide Description
In this work, we study existence, multiplicity, and concentration of positive solutions for the following class of quasilinear problems where is a N-function, ΔΦ is the Φ-Laplacian operator, ϵ is a positive parameter, V : ℝN → ℝ is a continuous function, and f : ℝ → ℝ is a C 1-function.
On effects of viscosity and magnetic fields on the largest growth rate of linear Rayleigh–Taylor instability57(2016); http://dx.doi.org/10.1063/1.4966924View Description Hide Description
In this article, we investigate the effect of viscosity on the largest growth rate in the linear Rayleigh–Taylor (RT) instability of a three-dimensional nonhomogeneous incompressible viscous flow in a bounded domain. By adapting a modified variational approach and careful analysis, we show that the largest growth rate in linear RT instability tends to zero as the viscosity coefficient goes to infinity. Moreover, the largest growth rate increasingly converges to one of the corresponding inviscid fluids as the viscosity coefficient goes to zero. Applying these analysis techniques to the corresponding viscous magnetohydrodynamic fluids, we can also show that the largest growth rate in linear magnetic RT instability tends to zero as the strength of horizontal (or vertical) magnetic field increasingly goes to a critical value.
57(2016); http://dx.doi.org/10.1063/1.4967299View Description Hide Description
Two kinds of occurrence mechanism on the phenomenon of concentration and the formation of delta shock wave in vanishing pressure limit of solutions to the modified Chaplygin gas equations are analyzed and identified. The Riemann problem of the modified Chaplygin gas equations is first solved. Then it is shown that, as the pressure vanishes, any two-shock Riemann solution tends to a delta-shock solution to the transport equations, and the intermediate density between the two shocks tends to a weighted δ-measure which forms a delta shock wave; any two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution to the transport equations, and the nonvacuum intermediate state in between tends to a vacuum state. It is also shown that, as the pressure approaches the generalized Chaplygin gas pressure, any two-shock Riemann solution tends to a delta-shock solution to the generalized Chaplygin gas equations. Some numerical results are presented to show the formation process of delta shock waves and vacuum states.
Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type with critical growth in ℝN57(2016); http://dx.doi.org/10.1063/1.4967976View Description Hide Description
In this paper, we deal with the existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type with critical nonlinearity: , (t, x) ∈ ℝ × ℝN. By using Lions’ second concentration-compactness principle and concentration-compactness principle at infinity to prove that (PS) condition holds locally and by variational method, we prove that it has at least one solution and for any m ∈ ℕ, it has at least m pairs of solutions.
57(2016); http://dx.doi.org/10.1063/1.4967975View Description Hide Description
In this paper, we study the compressible viscoelastic flows in three-dimensional whole space. Under the assumption of small initial data, we establish the unique global solution by the energy method. Furthermore, we obtain the time decay rates of the higher-order spatial derivatives of the solution if the initial data belong to L 1(ℝ3) additionally.
57(2016); http://dx.doi.org/10.1063/1.4967952View Description Hide Description
We propose a multi-component generalization of the modified short pulse (SP) equation which was derived recently as a reduction of Feng’s two-component SP equation. Above all, we address the two-component system in depth. We obtain the Lax pair, an infinite number of conservation laws and multisoliton solutions for the system, demonstrating its integrability. Subsequently, we show that the two-component system exhibits cusp solitons and breathers for which the detailed analysis is performed. Specifically, we explore the interaction process of two cusp solitons and derive the formula for the phase shift. While cusp solitons are singular solutions, smooth breather solutions are shown to exist, provided that the parameters characterizing the solutions satisfy certain conditions. Last, we discuss the relation between the proposed system and existing two-component SP equations.
- Representation Theory and Algebraic Methods
57(2016); http://dx.doi.org/10.1063/1.4966280View Description Hide Description
We describe a new technique to obtain representations of the braid group Bn from the ℜ–matrix of a quantum deformed algebra of the one dimensional harmonic oscillator. We consider the action of the ℜ–matrix not on the tensor product of representations of the algebra, that in the harmonic oscillator case are infinite dimensional, but on the subspace of the tensor product corresponding to the lowest weight vectors.
57(2016); http://dx.doi.org/10.1063/1.4966925View Description Hide Description
We find the ℓ-weights and the corresponding ℓ-weight vectors for the finite and infinite dimensional representations of the quantum loop algebras Uq(ℒ(𝔰𝔩2)) and Uq(ℒ(𝔰𝔩3)) obtained from the Verma representations of the quantum groups Uq(𝔤𝔩2) and Uq(𝔤𝔩3) via the Jimbo’s homomorphism. Then we find the ℓ-weights and the ℓ-weight vectors for the q-oscillator representations of the positive Borel subalgebras of the same quantum loop algebras. This allows, in particular, to relate the q-oscillator and prefundamental representations.
57(2016); http://dx.doi.org/10.1063/1.4967255View Description Hide Description
To a weighted graph can be associated a bipartite graph planar algebra . We construct and study the symmetric enveloping inclusion of . We show that this construction is equivariant with respect to the automorphism group of . We consider subgroups G of the automorphism of such that the G-fixed point space is a subfactor planar algebra. As an application we show that if G is amenable, then is amenable as a subfactor planar algebra. We define the notions of a cocycle action of a Hecke pair on a tracial von Neumann algebra and the corresponding crossed product. We show that a large class of symmetric enveloping inclusions of subfactor planar algebras can be described by such a crossed product.
- Many-Body and Condensed Matter Physics
57(2016); http://dx.doi.org/10.1063/1.4966642View Description Hide Description
We consider the 1/N-expansion of the moments of the proper delay times for a ballistic chaotic cavity supporting N scattering channels. In the random matrix approach, these moments correspond to traces of negative powers of Wishart matrices. For systems with and without broken time reversal symmetry (Dyson indices β = 1 and β = 2), we obtain a recursion relation, which efficiently generates the coefficients of the 1/N-expansion of the moments. The integrality of these coefficients and their possible diagrammatic interpretation is discussed.
- Quantum Mechanics
Trace formulas for the matrix Schrödinger operator on the half-line with general boundary conditions57(2016); http://dx.doi.org/10.1063/1.4964447View Description Hide Description
57(2016); http://dx.doi.org/10.1063/1.4964390View Description Hide Description
In this article we consider inverse eigenvalue problems for the Schrödinger operator on a finite interval. We extend and strengthen previously known uniqueness theorems. A partially known potential is identified by some sets of eigenvalues and norming constants.
- Quantum Information and Computation
There is no direct generalization of positive partial transpose criterion to the three-by-three case57(2016); http://dx.doi.org/10.1063/1.4966984View Description Hide Description
We show that there cannot exist a straightforward generalization of the famous positive partial transpose criterion to three-by-three systems. We call straightforward generalizations that use a finite set of positive maps and arbitrary local rotations of the tested two-partite state. In particular, we show that a family of extreme positive maps discussed in a paper by Ha and Kye [Open Syst. Inf. Dyn. 18, 323–337 (2011)], cannot be replaced by a finite set of witnesses in the task of entanglement detection in three-by-three systems. In a more mathematically elegant parlance, our result says that the convex cone of positive maps of the set of three-dimensional matrices into itself is not finitely generated as a mapping cone.
- Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory
57(2016); http://dx.doi.org/10.1063/1.4965446View Description Hide Description
We construct a family of constant curvature metrics on the Moyal plane and compute the Gauss–Bonnet term for each of them. They arise from the conformal rescaling of the metric in the orthonormal frame approach. We find a particular solution, which corresponds to the Fubini–Study metric and which equips the Moyal algebra with the geometry of a noncommutative sphere.
57(2016); http://dx.doi.org/10.1063/1.4966641View Description Hide Description
In this note we further develop the duality between supersymmetric gauge theories in various dimensions and elliptic integrable systems such as Ruijsenaars-Schneider model and periodic intermediate long wave hydrodynamics. These models arise in instanton counting problems and are described by certain elliptic algebras. We discuss the correspondence between the two types of models by employing the large-n limit of the dual gauge theory. In particular, we provide non-Abelian generalization of our previous result on the intermediate long wave model.
57(2016); http://dx.doi.org/10.1063/1.4967969View Description Hide Description
Coset methods are used to determine the action of a co-dimension one brane (domain wall) embedded in (d + 1)-dimensional AdS space in the Carroll limit in which the speed of light goes to zero. The action is invariant under the non-linearly realized symmetries of the AdS-Carroll spacetime. The Nambu-Goldstone field exhibits a static spatial distribution for the brane with a time varying momentum density related to the brane’s spatial shape as well as the AdS-C geometry. The AdS-C vector field dual theory is obtained.
- General Relativity and Gravitation
57(2016); http://dx.doi.org/10.1063/1.4966552View Description Hide Description
In this paper, we regularize the Kepler problem on κ-spacetime in several different ways. First, we perform a Moser-type regularization and then we proceed for the Ligon-Schaaf regularization to our problem. In particular, generalizing Heckman and de Laat [J. Symplectic Geom. 10, 463-473 (2012)] in the noncommutative context, we show that the Ligon-Schaaf regularization map following from an adaptation of the Moser regularization can be generalized to the Kepler problem on κ-spacetime.
57(2016); http://dx.doi.org/10.1063/1.4967954View Description Hide Description
In this paper we study finite time future singularities in anisotropic Bianchi type I models. It is shown that there exist future singularities similar to Big Rip ones (which appear in the framework of phantom Friedmann-Robertson-Walker cosmologies). Specifically, in an ellipsoidal anisotropic scenario or in a fully anisotropic scenario, the three directional and average scale factors may diverge at a finite future time, together with energy densities and anisotropic pressures. We call these singularities “Anisotropic Big Rip Singularities.” We show that there also exist Bianchi type I models filled with matter, where one or two directional scale factors may diverge. Another type of future anisotropic singularities is shown to be present in vacuum cosmologies, i.e., Kasner spacetimes. These singularities are induced by the shear scalar, which also blows up at a finite time. We call such a singularity “Vacuum Rip.” In this case one directional scale factor blows up, while the other two and average scale factors tend to zero.
57(2016); http://dx.doi.org/10.1063/1.4967951View Description Hide Description
We give the most general conditions to date which lead to uniqueness of the general relativistic Hamiltonian. Namely, we show that all spatially covariant generalizations of the scalar constraint which extend the standard one while remaining quadratic in the momenta are second class. Unlike previous investigations along these lines, we do not require a specific Poisson bracket algebra, and the quadratic dependence on the momenta is completely general, with an arbitrary local operator as the kinetic term.