arXiv subjects

Bjorn K. Berntson

Publications and source records attributed to Bjorn K. Berntson.

27 records · Page 2Linked to original sources

arXiv2020

Non-chiral Intermediate Long Wave equation and inter-edge effects in narrow quantum Hall systems

We present a non-chiral version of the Intermediate Long Wave (ILW) equation that can model nonlinear waves propagating on two opposite edges of a quantum Hall system, taking into account inter-edge interactions. We obtain exact soliton solutions governed by the hyperbolic Calogero-Moser-Sutherland (CMS) model, and we give a Lax pair, a Hirota form, and conservation laws for this new equation. We also present a periodic non-chiral ILW equation, together with its soliton solutions governed by the elliptic CMS model.

math-ph↗

arXiv2020

A new way to classify 2D higher order quantum superintegrable systems

We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symmetry operators of arbitrary order for the Schrödinger eigenvalue equation $HΨ\equiv (Δ_2 +V)Ψ=EΨ$ on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. We apply the method, as an example, to revisit the Tremblay and Winternitz (2010) derivation of the Painlevé VI potential for a 3rd order superintegrable flat space system that separates in polar coordinates and, as new results, we give a listing of the possible potentials on the 2-sphere that separate in spherical coordinates and 2-hyperbolic (two-sheet) potentials separating in horocyclic coordinates. In particular, we show that the Painlevé VI potential also appears for a 3rd order superintegrable system on the 2-sphere that separates in spherical coordinates, as well as a 3rd order superintegrable system on the 2-hyperboloid that separates in spherical coordinates and one that separates in horocyclic coordinates. Our aim is to develop tools for analysis and classification of higher order superintegrable systems on any 2D Riemannian space, not just Euclidean space.

math-ph↗

arXiv2020

Multi-solitons of the half-wave maps equation and Calogero-Moser spin-pole dynamics

We consider the half-wave maps (HWM) equation which provides a continuum description of the classical Haldane-Shastry spin chain on the real line. We present exact multi-soliton solutions of this equation. Our solutions describe solitary spin excitations that can move with different velocities and interact in a non-trivial way. We make an ansatz for the solution allowing for an arbitrary number of solitons, each described by a pole in the complex plane and a complex spin variable, and we show that the HWM equation is satisfied if these poles and spins evolve according to the dynamics of an exactly solvable spin Calogero-Moser (CM) system with certain constraints on initial conditions. We also find first order equations providing a Bäcklund transformation of this spin CM system, generalize our results to the periodic HWM equation, and provide plots that visualize our soliton solutions.

math-ph↗

arXiv2020

A new approach to analysis of 2D higher order quantum superintegrable systems

We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symmetry operators of arbitrary order for the Schrödinger eigenvalue equation $HΨ\equiv (Δ_2 +V)Ψ=EΨ$ on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. Most of this paper is devoted to describing the method. Details will be provided elsewhere. As examples we revisit the Tremblay and Winternitz derivation of the Painlevé VI potential for a 3rd order superintegrable flat space system that separates in polar coordinates and we show that the Painlevé VI potential also appears for a 3rd order superintegrable system on the 2-sphere that separates in spherical coordinates, as well as a 3rd order superintegrable system on the 2-hyperboloid that separates in spherical coordinates and one that separates in horocyclic coordinates. The purpose of this project is to develop tools for analysis and classification of higher order superintegrable systems on any 2D Riemannian space, not just Euclidean space.

math-ph↗

arXiv2018

Differential capacitance of ionic liquids according to lattice-gas mean-field model with nearest-neighbor interactions

The Bragg-Williams free energy is used to incorporate nearest-neighbor interactions into the lattice gas model of a solvent-free ionic liquid near a planar electrode. We calculate the differential capacitance from solutions of the mean-field consistency relation, arriving at an explicit expression in the limit of a weakly charged electrode. The two additional material parameters that appear in the theory -- the degree of nonideality and the resistance to concentration changes of each ion type -- give rise to different regimes that we identify and discuss. As the nonideality parameter, which becomes more positive for stronger nearest-neighbor attraction between like-charged ions, increases and the electrode is weakly charged, the differential capacitance is predicted to transition through a divergence and subsequently adopt negative values just before the ionic liquid becomes structurally unstable. This is associated with the spontaneous charging of an electrode at vanishing potential. The physical origin of the divergence and the negative sign of the differential capacitance is a nonmonotonic relationship between surface potential and surface charge density, which reflects the formation of layered domains alternatingly enriched in counterions and coions near the electrode. The decay length of this layered domain pattern, which can be many times larger than the ion size, is reminiscent of the recently introduced concept of "underscreening".

physics.chem-ph↗

arXiv2018

Debye-Hückel Theory of Weakly Curved Macroions: Implementing Ion Specificity through a Composite Coloumb-Yukawa Interaction Potential

The free energy of a weakly curved, isolated macroion embedded in a symmetric 1:1 electrolyte solution is calculated on the basis of linear Debye-Hückel theory, thereby accounting for non-electrostatic Yukawa pair interactions between the mobile ions and of the mobile ions with the macroion surface, present in addition to the electrostatic Coulomb potential. The Yukawa interactions between anion-anion, cation-cation, and anion-cation pairs are independent from each other and serve as a model for solvent-mediated ion-specific effects. We derive expressions for the free energy of a planar surface, the spontaneous curvature, the bending stiffness, and the Gaussian modulus. It is shown that a perturbation expansion, valid if the Yukawa interactions make a small contribution to the overall free energy, yields simple analytic results that exhibit good agreement with the general free energy over the range of experimentally relevant interaction parameters.

cond-mat.soft↗

arXiv2018

Curvature Elasticity of the Electric Double Layer

Mean-field electrostatics is used to calculate the bending moduli of an electric double layer for fixed surface charge density of a macroion in a symmetric 1:1 electrolyte. The resulting expressions for bending stiffness, Gaussian modulus, and spontaneous curvature refer to a general underlying equation of state of the electrolyte, subject to a local density approximation and the absence of dipole and higher-order fields. We present results for selected applications: the lattice-gas Poisson-Fermi model with and without asymmetric ion sizes, and the Poisson-Carnahan-Starling model.

cond-mat.soft↗

arXiv2018

Special Solutions of Bi-Riccati Delay-Differential Equations

Delay-differential equations are functional differential equations that involve shifts and derivatives with respect to a single independent variable. Some integrability candidates in this class have been identified by various means. For three of these equations we consider their elliptic and soliton-type solutions. Using Hirota's bilinear method, we find that two of our equations possess three-soliton-type solutions.

nlin.SI↗

arXiv2012

Observation of PT phase transition in a simple mechanical system

If a Hamiltonian is PT symmetric, there are two possibilities: Either the eigenvalues are entirely real, in which case the Hamiltonian is said to be in an unbroken-PT-symmetric phase, or else the eigenvalues are partly real and partly complex, in which case the Hamiltonian is said to be in a broken-PT-symmetric phase. As one varies the parameters of the Hamiltonian, one can pass through the phase transition that separates the unbroken and broken phases. This transition has recently been observed in a variety of laboratory experiments. This paper explains the phase transition in a simple and intuitive fashion and then describes an extremely elementary experiment in which the phase transition is easily observed.

math-ph↗