Mathematical Physics, Analysis and Geometry最新文献

We consider planar piecewise discontinuous differential systems formed by either linear centers or linear Hamiltonian saddles and separated by the algebraic curve (y=x^n) with (n ge 2). We provide in a very short way an upper bound of the number of limit cycles that these differential systems can have in terms of n, proving the extended 16th Hilbert problem in this case. In particular, we show that for (n=2) this bound can be reached.

In 2015, Vladimir Fock proved that the spectral transform, associating to an element of a dimer cluster integrable system its spectral data, is birational by constructing an inverse map using theta functions on Jacobians of spectral curves. We provide an alternate construction of the inverse map that involves only rational functions in the spectral data.

One of many manifestations of a deep relation between the topology of the moduli spaces of algebraic curves and the theory of integrable systems is a recent construction of Arsie, Lorenzoni, Rossi, and the first author associating an integrable system of evolutionary PDEs to an F-cohomological field theory (F-CohFT), which is a collection of cohomology classes on the moduli spaces of curves satisfying certain natural splitting properties. Typically, these PDEs have an infinite expansion in the dispersive parameter, which happens because they involve contributions from the moduli spaces of curves of arbitrarily large genus. In this paper, for each rank (Nge 2), we present a family of F-CohFTs without unit, for which the equations of the associated integrable system have a finite expansion in the dispersive parameter. For (N=2), we explicitly compute the primary flows of this integrable system.

In the present paper, we consider a p-adic Ising model on a Cayley tree. The existence of non-periodic p-adic generalized Gibbs measures of this model is investigated. In particular, we construct p-adic analogue of the Bleher–Ganikhodjaev construction and generalize some constructive methods. Moreover, the boundedness of obtained measures are established, which yields the occurrence of a phase transition.

The relative entropy of certain states on the algebra of canonical anticommutation relations (CAR) is studied in the present work. The CAR algebra is used to describe fermionic degrees of freedom in quantum mechanics and quantum field theory. The states for which the relative entropy is investigated are multi-excitation states (similar to multi-particle states) with respect to KMS states defined with respect to a time-evolution induced by a unitary dynamical group on the one-particle Hilbert space of the CAR algebra. If the KMS state is quasifree, the relative entropy of multi-excitation states can be explicitly calculated in terms of 2-point functions, which are defined entirely by the one-particle Hilbert space defining the CAR algebra and the Hamilton operator of the dynamical group on the one-particle Hilbert space. This applies also in the case that the one-particle Hilbert space Hamilton operator has a continuous spectrum so that the relative entropy of multi-excitation states cannot be defined in terms of von Neumann entropies. The results obtained here for the relative entropy of multi-excitation states on the CAR algebra can be viewed as counterparts of results for the relative entropy of coherent states on the algebra of canonical commutation relations which have appeared recently. It turns out to be useful to employ the setting of a self-dual CAR algebra introduced by Araki.

We study elliptic solutions of the recently introduced Toda lattice with the constraint of type B and derive equations of motion for their poles. The dynamics of poles is given by the deformed Ruijsenaars–Schneider system. We find its commutation representation in the form of the Manakov triple and study properties of the spectral curve. By studying more general elliptic solutions (elliptic families), we also suggest an extension of the deformed Ruijsenaars–Schneider system to a field theory.

We present a general algorithm constructing a discretization of a classical field theory from a Lagrangian. We prove a new discrete Noether theorem relating symmetries to conservation laws and an energy conservation theorem not based on any symmetry. This gives exact conservation laws for several theories, e.g., lattice electrodynamics and gauge theory. In particular, we construct a conserved discrete energy–momentum tensor, approximating the continuum one at least for free fields. The theory is stated in topological terms, such as coboundary and products of cochains.

In this paper, for the extended lattice Gelfand–Dickey hierarchy, we construct its n-fold Darboux transformation and additional flows. And we prove that these flows are actually symmetries of the extended lattice Gelfand–Dickey hierarchy. Further, we show how the additional flows act on the tau function. On this basis, we generalize the extended lattice Gelfand–Dickey hierarchy to the multicomponent and noncommutative versions, and give the Lax equations, Sato equations, zero-curvature equations and other equivalent expressions of these versions. Moreover, we investigate their Darboux transformations and additional symmetries.

For the Fröhlich model of the large polaron, we prove that the ground state energy as a function of the total momentum has a unique global minimum at momentum zero. This implies the non-existence of a ground state of the translation invariant Fröhlich Hamiltonian and thus excludes the possibility of a localization transition at finite coupling.

The two- and three-dimensional incompressible backward stochastic convective Brinkman–Forchheimer (BSCBF) equations on a torus driven by Lévy noise are considered in this paper. A-priori estimates for adapted solutions of the finite-dimensional approximation of 2D and 3D BSCBF equations are obtained. For a given terminal data, the existence and uniqueness of pathwise adapted strong solutions is proved by using a standard Galerkin (or spectral) approximation technique and exploiting the monotonicity arguments. We also establish the continuity of the adapted solutions with respect to the terminal data. The above results are obtained for the absorption exponent (rin [1,infty )) for (d=2) and (rin [3,infty )) for (d=3), and any Brinkman coefficient (mu >0), Forchheimer coefficient (beta >0), and hence the 3D critical case ((r=3)) is also handled successfully. We deduce analogous results for 2D backward stochastic Navier–Stokes equations perturbed by Lévy noise also.