Theoretical and Mathematical Physics最新文献

We consider a model of a circular network of neurons where the functioning of each neuron is described by an equation with two delays. The model under study is a modification considered in the paper of Glyzin et al., where the model of a solitary neuron is based on of the equation with one delay—the Hutchinson equation. We construct discrete traveling waves, i.e., a periodic solution of the system such that all components coincide with the same function shifted by a quantity that is multiple of a certain parameter. To find this solution, we study an auxiliary differential-difference equation of the Volterra type with three delays. For this equation, for any natural (m) and (n), we establish the existence of a periodic solution that contains (m) packets, each of which contains (n) bursts per period.

Motivated by the works published in 2003 by Chakraborty et al. [J. Operator Theory, 49 (2003), 185–201], and by Sakamoto and Tanimura [J. Math. Phys., 44 (2003), 5042–5069], we investigate the noncommutative (n)-torus in a magnetic field. We study the invariance of volume, integrated scalar curvature, and volume form using the method of perturbation by the inner derivation of the magnetic Laplacian in this geometric framework. Moreover, we derive the magnetic stochastic process describing the motion of a particle in a uniform magnetic field in this torus and deduce the properties of solutions of the corresponding magnetic quantum stochastic differential equation.

We introduce a new doubly unitary operator (isospectry) on wave functions, whose existence is equivalent to the equality of the spectra of two quantum billiards. It generates a map of nested billiards (induced reflection) with rich properties. This allows proving that in addition to isometry there is a unique realization of isospectrality of billiards, namely, a multivalued isometry. Then they are cellular and are constructed by reflections of the same cell. The algebra of their cellular subsets is isomorphic to a complete Boolean algebra and leads to the known generalized double negation formula, which corresponds to the logical foundations of quantum theory.

We discuss the general method of parameterization of matrix elements of local operators developed by Yu. M. Shirokov. This method is a core of one of the successful variants of the relativistic composite model developed by the authors, namely, the instant form of Dirac relativistic dynamics, which gives good results when describing composite quark and nucleon systems. Using the Shirokov parameterization, we construct operators of the electromagnetic current and the energy–momentum tensor of a composite system taking the Lorentz covariance and the conservation into account. As an example, we derive formulas for the electric and gravitational form factors of a pion.

We introduce a method for constructing negative symmetries from consistent triples of differential and differential-difference equations. We also study the relation between (3)D consistent equations in the discrete and continuous case.

We propose a forest fire model consisting of two equations, namely those describing the motion of the temperature front and the burned biomass front. To obtain a physically meaningful description of the solution behavior, we use equations with modular nonlinearity. For the proposed models, using asymptotic analysis methods, we have studied the existence of a solution in the form of a front. The asymptotic analysis allows us to estimate the speed of the front and determine the limits of the model applicability. When generalized to the two-dimensional case, the model can be used to simulate the motion of the combustion front in real forest fires, as well as to pose inverse problems for determining the amount of burned biomass after the passage of the combustion front.

We consider a class of systems of difference equations defined on an elementary quadrilateral of the (mathbb{Z}^2) lattice, define their eliminable and dynamical variables, and demonstrate their use. Using the existence of infinite hierarchies of symmetries as integrability criterion, we derive necessary integrability conditions and employ them in the construction of the lowest-order symmetries of a given system. These considerations are demonstrated with the help of three systems from the class of systems under consideration.

We consider solutions of a discrete Painlevé equation arising from a construction of quantum minimal surfaces by Arnlind, Hoppe, and Kontsevich, and in earlier work of Cornalba and Taylor on static membranes. While the discrete equation admits a continuum limit to the Painlevé I differential equation, we find that it has the same space of initial values as the Painlevé V equation with certain specific parameter values. We further explicitly show how each iteration of this discrete Painlevé I equation corresponds to a certain composition of Bäcklund transformations for Painlevé V, as was first remarked in a work by Tokihiro, Grammaticos, and Ramani. In addition, we show that some explicit special function solutions of Painlevé V, written in terms of modified Bessel functions, yield the unique positive solution of the initial value problem required for quantum minimal surfaces.

A brief history of the Birkhoff curve and Wada basins is presented. The Birkhoff curves are found to be indecomposable continua that are the common boundary of two regions having a single composant. Therefore, a Birkhoff curve contains at most one fixed point. A simplest geometric model of the Birkhoff curve has been constructed by matching the tails of the composants of the indecomposable Knaster continuum having two composants. By analogy to Knaster’s continuum, examples of indecomposable continua having four and and six composants are constructed. By pairwise matching the tails of composants, the indecomposable continua are obtained that are common boundaries of three and four regions, respectively. There exist two and four topologically different matchings, respectively. Clearly, (2n)-composant indecomposable continuum admits (2^n) ways of matching. These geometric constructions demonstrate the anatomical structure of nonwandering continua possessing the Wada property for a dynamical system acting on the plane with a single hyperbolic fixed point.

We present an exact analytical solution for the quantum dynamics of a charged particle subjected to both a time-dependent electric field and a static magnetic field aligned along the (z)-direction. Using a systematic approach based on successive unitary transformations, we reduce the original three-dimensional problem to a two-dimensional system of decoupled, time-dependent harmonic oscillators. This technique produces free parameters that allow us to impose constrains to derive the exact solution of Schrödinger equation with the time-dependent Hamiltonian through the explicit derivation of the quantum propagator and shows the equivalence of this approach to established path integral methods for such systems. The developed framework provides new insights into quantum systems with time-dependent electromagnetic fields and offers analytical solutions.