Volume 55, Issue 5, May 2014

A magnetic field is defined by the property that its divergence is zero in a threedimensional oriented Riemannian manifold. Each magnetic field generates a magnetic flow whose trajectories are curves called as magnetic curves. In this paper, we give a new variational approach to study the magnetic flow associated with the Killing magnetic field in a threedimensional oriented Riemann manifold, (M ^{3}, g). And then, we investigate the trajectories of the magnetic fields called as Nmagnetic and Bmagnetic curves.
 ARTICLES

 Partial Differential Equations

Nodal soliton solutions for generalized quasilinear Schrödinger equations
View Description Hide DescriptionThis paper is concerned with constructing nodal radial solutions for generalized quasilinear Schrödinger equations in which arise from plasma physics, fluid mechanics, as well as highpower ultashort laser in matter. For any given integer k ⩾ 0, by using a change of variables and minimization argument, we obtain a signchanging minimizer with k nodes of a minimization problem.

An unconstrained Lagrangian formulation and conservation laws for the Schrödinger map system
View Description Hide DescriptionWe consider energycritical Schrödinger maps from into and . Viewing such maps with respect to orthonormal frames on the pullback bundle provides a gauge field formulation of the evolution. We show that this gauge field system is the set of EulerLagrange equations corresponding to an action that includes a ChernSimons term. We also introduce the stressenergy tensor and derive conservation laws. In conclusion we offer comparisons between Schrödinger maps and the closely related ChernSimonsSchrödinger system.

The charge conserving PoissonBoltzmann equations: Existence, uniqueness, and maximum principle
View Description Hide DescriptionThe present article is concerned with the charge conserving PoissonBoltzmann (CCPB) equation in highdimensional bounded smooth domains. The CCPB equation is a PoissonBoltzmann type of equation with nonlocal coefficients. First, under the Robin boundary condition, we get the existence of weak solutions to this equation. The main approach is variational, based on minimization of a logarithmtype energy functional. To deal with the regularity of weak solutions, we establish a maximum modulus estimate for the standard PoissonBoltzmann (PB) equation to show that weak solutions of the CCPB equation are essentially bounded. Then the classical solutions follow from the elliptic regularity theorem. Second, a maximum principle for the CCPB equation is established. In particular, we show that in the case of global electroneutrality, the solution achieves both its maximum and minimum values at the boundary. However, in the case of global nonelectroneutrality, the solution may attain its maximum value at an interior point. In addition, under certain conditions on the boundary, we show that the global nonelectroneutrality implies pointwise nonelectroneutrality.

Existence and Regularity of Invariant Measures for the Three Dimensional Stochastic Primitive Equations
View Description Hide DescriptionWe establish the continuity of the Markovian semigroup associated with strong solutions of the stochastic 3D Primitive Equations, and prove the existence of an invariant measure. The proof is based on new moment bounds for strong solutions. The invariant measure is supported on strong solutions and is furthermore shown to have higher regularity properties.

Regularity criteria and uniform estimates for the Boussinesq system with temperaturedependent viscosity and thermal diffusivity
View Description Hide DescriptionIn this paper, we establish some regularity criteria for the 3D Boussinesq system with the temperaturedependent viscosity and thermal diffusivity. We also obtain some uniform estimates for the corresponding 2D case when the fluid viscosity coefficient is a positive constant.
 Quantum Mechanics

Nonhomogeneous solutions of a Coulomb Schrödinger equation as basis set for scattering problems
View Description Hide DescriptionWe introduce and study twobody Quasi Sturmian functions which are proposed as basis functions for applications in threebody scattering problems. They are solutions of a twobody nonhomogeneous Schrödinger equation. We present different analytic expressions, including asymptotic behaviors, for the pure Coulomb potential with a driven term involving either Slatertype or Laguerretype orbitals. The efficiency of Quasi Sturmian functions as basis set is numerically illustrated through a twobody scattering problem.

On the solvability of the generalized hyperbolic doublewell models
View Description Hide DescriptionWe present exact solutions for the Schrödinger equation with the hyperbolic doublewell potential . We show that the model preserves a finite dimensional polynomial space for some p and q. Thus using the Bethe ansatz method, we obtain closed form expressions for the spectrum and wavefunction, as well as the allowed parameter for the class , which is contrary to a report in a recent article [C. A. Downing, J. Math. Phys.54, 072101 (2013)]. We also discuss the hidden sl 2 algebraic structure of the class.

Selfadjointness and boundedness in quadratic quantization
View Description Hide DescriptionWe construct a counter example showing, for the quadratic quantization, the identity (Γ(T))* = Γ(T*) is not necessarily true. We characterize all operators on the oneparticle algebra whose quadratic quantization are selfadjoint operators on the quadratic Fock space. Finally, we discuss the boundedness of the quadratic quantization.

The fundamental gap for a class of Schrödinger operators on path and hypercube graphs
View Description Hide DescriptionWe consider the difference between the two lowest eigenvalues (the fundamental gap) of a Schrödinger operator acting on a class of graphs. In particular, we derive tight bounds for the gap of Schrödinger operators with convex potentials acting on the path graph. Additionally, for the hypercube graph, we derive a tight bound for the gap of Schrödinger operators with convex potentials dependent only upon vertex Hamming weight. Our proof makes use of tools from the literature of the fundamental gap theorem as proved in the continuum combined with techniques unique to the discrete case. We prove the tight bound for the hypercube graph as a corollary to our path graph results.
 Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

Perturbative description of the fermionic projector: Normalization, causality, and Furry's theorem
View Description Hide DescriptionThe causal perturbation expansion of the fermionic projector is performed with a contour integral method. Different normalization conditions are analyzed. It is shown that the corresponding lightcone expansions are causal in the sense that they only involve bounded line integrals. For the resulting loop diagrams we prove a generalized Furry theorem.

mechanics with diverse (4, 4, 0) multiplets: Explicit examples of hyperKähler with torsion, Clifford Kähler with torsion, and octonionic Kähler with torsion geometries
View Description Hide DescriptionWe present simple models of supersymmetric mechanics with ordinary and mirror linear (4, 4, 0) multiplets that give a transparent description of HyperKähler with Torsion (HKT), Clifford Kähler with Torsion (CKT), and Octonionic Kähler with Torsion (OKT) geometries. These models are treated in the and superfield approaches, as well as in the component approach. Our study makes manifest that the CKT and OKT supersymmetric sigma models are distinguished from the more simple HKT models by the presence of extra holomorphic torsion terms in the supercharges.

Emergence of complex and spinor wave functions in scale relativity. II. Lorentz invariance and bispinors
View Description Hide DescriptionOwing to the nondifferentiable nature of the theory of Scale Relativity, the emergence of complex wave functions, then of spinors and bispinors occurs naturally in its framework. The wave function is here a manifestation of the velocity field of geodesics of a continuous and nondifferentiable (therefore fractal) spacetime. In a first paper (Paper I), we have presented the general argument which leads to this result using an elaborate and more detailed derivation than previously displayed. We have therefore been able to show how the complex wave function emerges naturally from the doubling of the velocity field and to revisit the derivation of the nonrelativistic Schrödinger equation of motion. In the present paper (Paper II), we deal with relativistic motion and detail the natural emergence of the bispinors from such first principles of the theory. Moreover, while Lorentz invariance has been up to now inferred from mathematical results obtained in stochastic mechanics, we display here a new and detailed derivation of the way one can obtain a Lorentz invariant expression for the expectation value of the product of two independent fractal fluctuation fields in the sole framework of the theory of Scale Relativity. These new results allow us to enhance the robustness of our derivation of the two main equations of motion of relativistic quantum mechanics (the KleinGordon and Dirac equations) which we revisit here at length.
 Dynamical Systems

Stochastic dynamics of a delayed bistable system with multiplicative noise
View Description Hide DescriptionIn this paper we investigate the properties of a delayed bistable system under the effect of multiplicative noise. We first prove the existence and uniqueness of the positive solution and show that its moments are uniformly bounded. Then, we study stochastic dynamics of the solution in long time, the lower and upper bounds for the paths and an estimate for the average value are provided.

Phase portraits analysis of a barothropic system: The initial value problem
View Description Hide DescriptionIn this paper, we investigate the phase portraits features of a barothropic relaxing medium under pressure perturbations. In the starting point, we show within a thirdorder of accuracy that the previous system is modeled by a “dissipative” cubic nonlinear evolution equation. Paying particular attention to highfrequency perturbations of the system, we solve the initial value problem of the system both analytically and numerically while unveiling the existence of localized multivalued waveguide channels. Accordingly, we find that the “dissipative” term with a “dissipative” parameter less than some limit value does not destroy the ambiguous solutions. We address some physical implications of the results obtained previously.

Emergent behaviors of a holonomic particle system on a sphere
View Description Hide DescriptionWe study sufficient conditions for the asymptotic emergence of synchronous behaviors in a holonomic particle system on a sphere, which was recently introduced by Lohe [“NonAbelian Kuramoto model and synchronization,” J. Phys. A: Math. Theor.42, 395101–395126 (2009)]. These conditions depend only on the coupling strength and initial position diameter. For identical particles, we show that the position diameter approaches zero asymptotically under these sufficient conditions, i.e., all particles approach to the same position. For nonidentical particles, the particle positions do not shrink to one point, but can be squeezed into some small region whose diameter is inversely proportional to the coupling strength, when the coupling strength is large. We also provide several numerical results to confirm our analytical findings.
 Classical Mechanics and Classical Fields

Canonical transformations for hyperkahler structures and hyperhamiltonian dynamics
View Description Hide DescriptionWe discuss generalizations of the well known concept of canonical transformations for symplectic structures to the case of hyperkahler structures. Different characterizations, which are equivalent in the symplectic case, give rise to nonequivalent notions in the hyperkahler framework; we will thus distinguish between hyperkahler and canonical transformations. We also discuss the properties of hyperhamiltonian dynamics in this respect.
 Statistical Physics

Rigorous investigation of the reduced density matrix for the ideal Bose gas in harmonic traps by a loopgaslike approach
View Description Hide DescriptionIn this paper, we rigorously investigate the reduced density matrix (RDM) associated to the ideal Bose gas in harmonic traps. We present a method based on a sumdecomposition of the RDM allowing to treat not only the isotropic trap, but also general anisotropic traps. When focusing on the isotropic trap, the method is analogous to the loopgas approach developed by Mullin [“The loopgas approach to BoseEinstein condensation for trapped particles,” Am. J. Phys.68(2), 120 (2000)]. Turning to the case of anisotropic traps, we examine the RDM for some anisotropic trap models corresponding to some quasi1D and quasi2D regimes. For such models, we bring out an additional contribution in the local density of particles which arises from the mesoscopic loops. The close connection with the occurrence of generalizedBoseEinstein condensation is discussed. Our loopgaslike approach provides relevant information which can help guide numerical investigations on highly anisotropic systems based on the Path Integral Monte Carlo method.
 Methods of Mathematical Physics

A new approach for magnetic curves in 3D Riemannian manifolds
View Description Hide DescriptionA magnetic field is defined by the property that its divergence is zero in a threedimensional oriented Riemannian manifold. Each magnetic field generates a magnetic flow whose trajectories are curves called as magnetic curves. In this paper, we give a new variational approach to study the magnetic flow associated with the Killing magnetic field in a threedimensional oriented Riemann manifold, (M ^{3}, g). And then, we investigate the trajectories of the magnetic fields called as Nmagnetic and Bmagnetic curves.

Nonexistence of the usual scattering states for the generalized OstrovskyHunter equation
View Description Hide DescriptionWe consider the Cauchy problem for the generalized OstrovskyHunter equation u tx = u + (u^{ρ−1} u) xx . We prove the nonexistence of the usual scattering states.

Strongly asymmetric discrete Painlevé equations: The additive case
View Description Hide DescriptionWe examine a class of discrete Painlevé equations which present a strong asymmetry. These equations can be written as a system of two equations, the righthandsides of which do not have the same functional form. We limit here our investigation to two canonical families of the QuispelRobertsThompson (QRT) classification both of which lead to difference equations. Several new integrable discrete systems are identified.