Volume 57, Issue 2, February 2016
 ARTICLES

 Partial Differential Equations

Decay rates to viscous contact waves for the compressible NavierStokes equations
View Description Hide DescriptionIn this paper, we study the largetime asymptotic behavior of contact wave for the Cauchy problem of onedimensional compressible NavierStokes equations with zero viscosity. When the Riemann problem for the Euler system admits a contact discontinuity solution, we can construct a contact wave, which approximates the contact discontinuity on any finitetime interval for small heat conduction and then runs away from it for large time, and proves that it is nonlinearly stable provided that the strength of the contact discontinuity and the perturbation of the initial data are suitably small.

Landau damping in relativistic plasmas
View Description Hide DescriptionWe examine the phenomenon of Landau damping in relativistic plasmas via a study of the relativistic VlasovPoisson (rVP) system on the torus for initial data sufficiently close to a spatially uniform steady state. We find that if the steady state is regular enough (essentially in a Gevrey class of degree in a specified range) and if the deviation of the initial data from this steady state is small enough in a certain norm, the evolution of the system is such that its spatial density approaches a uniform constant value quasiexponentially fast (i.e., like for ). We take as a priori assumptions that solutions launched by such initial data exist for all times (by no means guaranteed with rVP, but a reasonable assumption since we are close to a spatially uniform state) and that the various norms in question are continuous in time (which should be a consequence of an abstract version of the CauchyKovalevskaya theorem). In addition, we must assume a kind of “reverse Poincaré inequality” on the Fourier transform of the solution. In spirit, this assumption amounts to the requirement that there exists 0 < ϰ < 1 so that the mass in the annulus for the solution launched by the initial data is uniformly small for all t. Typical velocity bounds for solutions to rVP launched by small initial data (at least on ℝ^{6}) imply this bound. We note that none of our results require spherical symmetry (a crucial assumption for many current results on rVP).

Energy conservative solutions to the system of full variational sineGordon equations in a unit sphere
View Description Hide DescriptionWe establish the global existence of a conservative weak solution to the Cauchy problem for a complete system of variational sineGordon equations, which models the motion of long waves on a neutral dipole chain in the continuum limit in a unit sphere. Although singularities may develop in finite time, the energy of the solution is conserved across singular times. We also obtain the continuous dependence of solutions on the given initial data.

An integral formula adapted to different boundary conditions for arbitrarily highdimensional nonlinear KleinGordon equations with its applications
View Description Hide DescriptionIn this paper, we are concerned with the initial boundary value problem of arbitrarily highdimensional KleinGordon equations, posed on a bounded domain Ω ⊂ ℝ^{d} for d ≥ 1 and equipped with the requirement of boundary conditions. We derive and analyze an integral formula which is proved to be adapted to different boundary conditions for general KleinGordon equations in arbitrarily highdimensional spaces. The formula gives a closedform solution to arbitrarily highdimensional homogeneous linear KleinGordon equations, which is totally different from the wellknown D’Alembert, Poisson, and Kirchhoff formulas. Some applications are included as well.

Twoloop renormalization group flow for Lorentzian manifolds
View Description Hide DescriptionWe study twoloop renormalization group flow (RG2) on Lorentzian threemanifolds. The fixed points of flow are analysed on a threedimensional Lorentzian manifold and we present a classification of the solitons that evolves only by homotheties. Furthermore, we prove the existence of a Lorentzian RG2 cigar soliton of constant positive curvature.

No blowup to a variational wave equation in liquid crystals
View Description Hide DescriptionWe establish the global existence of smooth solutions to the Cauchy problem for a system of variational wave equations in one space dimension modeling a type of nematic liquid crystals that has equal splay and bend coefficients.

Global small solutions to a tropical climate model without thermal diffusion
View Description Hide DescriptionWe obtain the global wellposedness of classical solutions to a tropical climate model with only the dissipation of the first baroclinic model of the velocity (−ηΔv) under small initial data. This model is a modified version of the original system derived by FriersonMajdaPauluis in Frierson et al. [Commun. Math. Sci. 2, 591626 (2004)]. The main difficulty is the absence of thermal diffusion. To overcome it, we exploit the structure of the equations coming from the coupled terms, dissipation term, and damp term. Then, we find the hidden thermal diffusion. In addition, based on the LittlewoodPaley theory, we establish a generalized commutator estimate, which may be applied to other partial differential equations.

Dressing method and quadratic bundles related to symmetric spaces. Vanishing boundary conditions
View Description Hide DescriptionWe consider quadratic bundles related to Hermitian symmetric spaces of the type SU(m + n)/S(U(m) × U(n)). The simplest representative of the corresponding integrable hierarchy is given by a multicomponent KaupNewell derivative nonlinear Schrödinger equation which serves as a motivational example for our general considerations. We extensively discuss how one can apply ZakharovShabat’s dressing procedure to derive reflectionless potentials obeying zero boundary conditions. Those could be used for one to construct fast decaying solutions to any nonlinear equation belonging to the same hierarchy. One can distinguish between generic soliton type solutions and rational solutions.

Energy decay of solutions for a variablecoefficient viscoelastic wave equation with a weak nonlinear dissipation
View Description Hide DescriptionIn this paper, we study decay properties of the solutions for a variablecoefficient viscoelastic wave equation with a weak nonlinear dissipative term. The method we used are piecewise multiplier methods and geometric analysis.
 Representation Theory and Algebraic Methods

Matrix exponentials, SU(N) group elements, and real polynomial roots
View Description Hide DescriptionThe exponential of an N × N matrix can always be expressed as a matrix polynomial of order N − 1. In particular, a general group element for the fundamental representation of SU(N) can be expressed as a matrix polynomial of order N − 1 in a traceless N × N hermitian generating matrix, with polynomial coefficients consisting of elementary trigonometric functions dependent on N − 2 invariants in addition to the group parameter. These invariants are just angles determined by the direction of a real Nvector whose components are the eigenvalues of the hermitian matrix. Equivalently, the eigenvalues are given by projecting the vertices of an simplex onto a particular axis passing through the center of the simplex. The orientation of the simplex relative to this axis determines the angular invariants and hence the real eigenvalues of the matrix.

Radial Bargmann representation for the Fock space of type B
View Description Hide DescriptionLet να,q be the probability and orthogonality measure for the qMeixnerPollaczek orthogonal polynomials, which has appeared in the work of Bożejko, Ejsmont, and Hasebe [J. Funct. Anal. 269, 1769–1795 (2015)] as the distribution of the (α, q)Gaussian process (the Gaussian process of type B) over the (α, q)Fock space (the Fock space of type B). The main purpose of this paper is to find the radial Bargmann representation of να,q. Our main results cover not only the representation of qGaussian distribution by van Leeuwen and Maassen [J. Math. Phys. 36, 4743–4756 (1995)] but also of q^{2}Gaussian and symmetric free Meixner distributions on ℝ. In addition, nontrivial commutation relations satisfied by (α, q)operators are presented.

Classical affine Walgebras associated to Lie superalgebras
View Description Hide DescriptionIn this paper, we prove classical affine Walgebras associated to Lie superalgebras (Wsuperalgebras), which can be constructed in two different ways: via affine classical Hamiltonian reductions and via taking quasiclassical limits of quantum affine Wsuperalgebras. Also, we show that a classical finite Wsuperalgebra can be obtained by a Zhu algebra of a classical affine Wsuperalgebra. Using the definition by Hamiltonian reductions, we find free generators of a classical Wsuperalgebra associated to a minimal nilpotent. Moreover, we compute generators of the classical Walgebra associated to spo(23) and its principal nilpotent. In the last part of this paper, we introduce a generalization of classical affine Wsuperalgebras called classical affine fractional Wsuperalgebras. We show these have Poisson vertex algebra structures and find generators of a fractional Wsuperalgebra associated to a minimal nilpotent.

The number radial coherent states for the generalized MICZKepler problem
View Description Hide DescriptionWe study the radial part of the McIntoshCisnerosZwanziger (MICZ)Kepler problem in an algebraic way by using the 𝔰𝔲(1, 1) Lie algebra. We obtain the energy spectrum and the eigenfunctions of this problem from the 𝔰𝔲(1, 1) theory of unitary representations and the tilting transformation to the stationary Schrödinger equation. We construct the physical Perelomov number coherent states for this problem and compute some expectation values. Also, we obtain the time evolution of these coherent states.

On Wigner transforms in infinite dimensions
View Description Hide DescriptionWe investigate the Schrödinger representations of certain infinitedimensional Heisenberg groups, using their corresponding Wigner transforms.
 ManyBody and Condensed Matter Physics

On the optical properties of carbon nanotubes. Part I. A general formula for the dynamical optical conductivity
View Description Hide DescriptionThis paper is the first one in a series of two articles in which we revisit the optical properties of singlewalled carbon nanotubes (SWNTs). Produced by rolling up a graphene sheet, SWNTs owe their intriguing properties to their cylindrical quasionedimensional (quasi1D) structure (the ratio length/radius is experimentally of order of 10^{3}). We model SWNT by circular cylinders of small diameters on the surface of which the conduction electron gas is confined by the electric field generated by the fixed carbon ions. The pairinteraction potential considered is the 3D Coulomb potential restricted to the cylinder. To reflect the quasi1D structure, we introduce a 1D effective manybody Hamiltonian which is the startingpoint of our analysis. To investigate the optical properties, we consider a perturbation by a uniform timedependent electric field modeling an incident light beam along the longitudinal direction. By using Kubo’s method, we derive within the linear response theory an asymptotic expansion in the lowtemperature regime for the dynamical optical conductivity at fixed density of particles. The leading term only involves the eigenvalues and associated eigenfunctions of the (unperturbed) 1D effective manybody Hamiltonian and allows us to account for the sharp peaks observed in the optical absorption spectrum of SWNT.
 Quantum Mechanics

On absence of bound states for weakly attractive δ′interactions supported on nonclosed curves in ℝ^{2}
View Description Hide DescriptionLet Λ ⊂ ℝ^{2} be a nonclosed piecewiseC^{1} curve, which is either bounded with two free endpoints or unbounded with one free endpoint. Let u±Λ ∈ L^{2}(Λ) be the traces of a function u in the Sobolev space H^{1}(ℝ^{2}∖Λ) onto two faces of Λ. We prove that for a wide class of shapes of Λ the Schrödinger operator with δ′interaction supported on Λ of strength ω ∈ L^{∞}(Λ; ℝ) associated with the quadratic form has no negative spectrum provided that ω is pointwise majorized by a strictly positive function explicitly expressed in terms of Λ. If, additionally, the domain ℝ^{2}∖Λ is quasiconical, we show that . For a bounded curve Λ in our class and nonvarying interaction strength ω ∈ ℝ, we derive existence of a constant ω∗ > 0 such that for all ω ∈ (−∞, ω∗]; informally speaking, bound states are absent in the weak coupling regime.

Timedependent pseudofermionic systems and coherent states
View Description Hide DescriptionWe show, by means of similarity transformations, that the timedependent fermionic systems are associated to the timedependent pseudofermionic systems. A general construction of time dependent fermionic coherent states (FCSs) describing the twolevel dissipative system driven by a periodic electromagnetic field is developed, and a strict parallelism between FCS and the time dependent pseudofermionic coherent states (PFCSs) is established and examined. We discuss properties of the constructed FCS and PFCS.

Green’s functions and energy eigenvalues for deltaperturbed spacefractional quantum systems
View Description Hide DescriptionStarting from the propagator, we introduced a timeordered perturbation expansion and employed Wick rotation to obtain a general energydependent Green’s function expressions for spacefractional quantum systems with Dirac deltafunction perturbation. We then obtained the Green’s functions and equations for the bound state energies for the spacefractional Schrödinger equation with single and double Dirac delta well potentials and the deltaperturbed infinite well.

On the existence of bound states in asymmetric leaky wires
View Description Hide DescriptionWe analyze spectral properties of a leaky wire model with a potential bias. It describes a twodimensional quantum particle exposed to a potential consisting of two parts. One is an attractive δinteraction supported by a nonstraight, piecewise smooth curve dividing the plane into two regions of which one, the “interior,” is convex. The other interaction component is a constant positive potential V0 in one of the regions. We show that in the critical case, V0 = α^{2}, the discrete spectrum is nonvoid if and only if the bias is supported in the interior. We also analyze the noncritical situations, in particular, we show that in the subcritical case, V0 < α^{2}, the system may have any finite number of bound states provided the angle between the asymptotes of is small enough.

Dynamical error bounds for continuum discretisation via Gauss quadrature rules—A LiebRobinson bound approach
View Description Hide DescriptionInstances of discrete quantum systems coupled to a continuum of oscillators are ubiquitous in physics. Often the continua are approximated by a discrete set of modes. We derive error bounds on expectation values of system observables that have been time evolved under such discretised Hamiltonians. These bounds take on the form of a function of time and the number of discrete modes, where the discrete modes are chosen according to Gauss quadrature rules. The derivation makes use of tools from the field of LiebRobinson bounds and the theory of orthonormal polynomials.