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Theoretical and Mathematical Physics
Journal Prestige (SJR): 0.409  Citation Impact (citeScore): 1 Number of Followers: 8  Hybrid journal (It can contain Open Access articles)ISSN (Print) 1573-9333 - ISSN (Online) 0040-5779 Published by Springer-Verlag  [2468 journals]
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- Geometry and probability on the noncommutative 2-torus in a magnetic field
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Abstract: We describe the geometric and probabilistic properties of a noncommutative \(2\) -torus in a magnetic field. We study the volume invariance, integrated scalar curvature, and the volume form by using the operator method of perturbation by an inner derivation of the magnetic Laplacian operator on the noncommutative \(2\) -torus. We then analyze the magnetic stochastic process describing the motion of a particle subject to a uniform magnetic field on the noncommutative \(2\) -torus, and discuss the related main properties.
PubDate: 2024-08-01
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- Asymptotics of solutions of the Cauchy problem for a singularly perturbed
operator differential transport equation-
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Abstract: We consider singularly perturbed operator differential transport equations of a special form in the case where the transport operator acts on space–time variables; a linear operator acting on an additional variable describes the interaction that “scrambles” the solution with respect to that variable. We construct a formal asymptotic expansion of the solution of the Cauchy problem for a singularly perturbed operator differential transport equation with small nonlinearity and weak diffusion in the case of several spatial variables. Under some conditions assumed for these problems, the leading term of the asymptotics is described by a quasilinear parabolic equation. The remainder term is estimated with respect to the residual under certain conditions.
PubDate: 2024-08-01
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- Dynamical properties of a diffusion-coupled system of differential
equations with an additional internal coupling-
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Abstract: We study the dynamics of a system of differential equations with the diffusion interaction and an additional internal coupling. Such systems are interesting because a slight variation in the coefficient at the additional coupling allows obtaining intricate scenarios of phase rearrangements. For the system under study, we find the critical dependence of the parameters such that zero equilibrium loses stability as two spatially inhomogeneous states appear in one case and a cycle in another case. With the parameter values close to the critical ones, asymptotic formulas are obtained for the regimes that branch off from the zero solution.
PubDate: 2024-08-01
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- Periodic solutions of a differential equation with a discontinuous delayed
neutral-type feedback-
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Abstract: We consider a differential equation with a discontinuous delayed neutral-type feedback. In the phase space, we describe classes of initial functions that depend on a number of parameters. We show that in a certain time, solutions return to an analogous class, possibly with other parameters. The analysis of the change in the parameters allows describing periodic solutions and their stability. We show that infinitely many of stable periodic solutions exist.
PubDate: 2024-08-01
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- Modeling the traffic flow in areas with different speed limits
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Abstract: The main result of this paper is a mathematical model that describes the dynamics of the motion of several cars in areas with different speed limits. As such areas, we can consider speed limit zones and speed bumps or uneven road surfaces. The model is a system of differential equations with a delayed argument. The dynamical properties of the model are studied by numerical methods. A computer program has been developed that uses the model to describe the motion of traffic flows in various road situations. The simulation results coincide with the observation data of real traffic flows.
PubDate: 2024-08-01
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- Generalized Chaos game in an extended hyperbolic plane
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Abstract: We propose and theoretically substantiate an algorithm for conducting a generalized Chaos game with an arbitrary jump on finite convex polygons of the extended hyperbolic plane \(H^2\) whose components in the Cayley–Klein projective model are the Lobachevsky plane and its ideal domain. In particular, the defining identities for a point dividing an elliptic, hyperbolic, or parabolic segment in a given ratio are proved, and formulas for calculating the coordinates of such a point at a canonical frame of the first type are obtained. The results of a generalized Chaos game conducted using the advanced software package pyv are presented.
PubDate: 2024-08-01
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- Kramers–Wannier duality and Tutte polynomials
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Abstract: We study applications of the connection between the partition functions of the Potts models and Tutte polynomials: it is demonstrated how the Kramers–Wannier duality can be derived from the Tutte duality. Using the “contraction–elimination” relation and the Biggs formalism, we derive the high-temperature expansion and discuss possible methods for generalizing the Kramers–Wannier duality to models on nonplanar graphs.
PubDate: 2024-08-01
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- Analysis of the asymptotic convergence of periodic solution of the
Mackey–Glass equation to the solution of the limit relay equation-
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Abstract: The relaxation self-oscillations of the Mackey–Glass equation are studied under the assumption that the exponent in the nonlinearity denominator is a large parameter. We consider the case where the limit relay equation, which arises as the large parameter tends to infinity, has a periodic solution with the smallest number of breaking points on the period. In this case, we prove the existence of a periodic solution of the Mackey–Glass equation that is asymptotically close to the periodic solution of the limit equation.
PubDate: 2024-08-01
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- Nonlinear waves in a parabolic equation with a spatial argument rescaling
operator and with time delay-
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Abstract: We study bifurcations of nonlinear waves (spatially inhomogeneous solutions) emerging from homogeneous equilibrium states of an initial boundary value problem, arising in nonlinear optics, for a nonlinear parabolic equation on a disk with a spatial argument rescaling operator and with time delay. In the plane of the main parameters of the equation, we construct stability (instability) domains of homogeneous equilibrium states and study the dynamics of the stability domains depending on the rescaling coefficient. We investigate the mechanisms of stability loss by homogeneous equilibrium states, the possible bifurcations of spatially inhomogeneous self-oscillatory solutions, and their stability. We demonstrate the possibility of bifurcation of stable rotational and spiral waves.
PubDate: 2024-08-01
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- Stationary thermal front in the problem of reconstructing the
semiconductor thermal conductivity coefficient using simulation data-
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Abstract: We study the problem of the existence of stationary, asymptotically Lyapunov-stable solutions with internal transition layers in nonlinear heat conductance problems with a thermal flow containing a negative exponent. We formulate sufficient conditions for the existence of classical solutions with internal layers in such problems. We construct an asymptotic approximation of an arbitrary-order for the solution with a transition layer. We substantiate the algorithm for constructing the formal asymptotics and study the asymptotic Lyapunov stability of the stationary solution with an internal layer as a solution of the corresponding parabolic problem with the description of the local attraction domain of the stable stationary solution. As an application, we present a new effective method for reconstructing the nonlinear thermal conductivity coefficient with a negative exponent using the position of the stationary thermal front in combination with observation data.
PubDate: 2024-08-01
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- Second-order quantum argument shifts in $$Ugl_d$$
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Abstract: We describe an explicit formula for the second-order quantum argument shifts of an arbitrary central element of the universal enveloping algebra of a general linear Lie algebra. We identify the generators of the subalgebra generated by the quantum argument shifts up to the second order.
PubDate: 2024-08-01
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- Mechanism for the formation of an inhomogeneous nanorelief and
bifurcations in a nonlocal erosion equation-
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Abstract: We continue studies of the nonlocal erosion equation that is used as a mathematical model of the formation of a spatially inhomogeneous relief on semiconductor surfaces. We show that such a relief can form as a result of local bifurcations in the case where the stability of the spatially homogeneous equilibrium state changes. We consider a periodic boundary-value problem and study its codimension- \(2\) bifurcations. For solutions describing an inhomogeneous relief, we obtain asymptotic formulas and study their stability. The analysis of the mathematical problem is based on modern methods of the theory of dynamical systems with an infinite-dimensional phase space, in particular, on the method of integral manifolds and on the theory of normal forms.
PubDate: 2024-07-01
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- On contrast structures in a problem of the baretting effect theory
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Abstract: We obtain a contrast-structure type solution of a system of equations for the baretting effect that include a nonlinear singularly perturbed parabolic equation and an additional nonlocal integral relation. We prove the existence of the solution with an internal transition layer and construct the asymptotic approximation of this solution. We obtain estimates of the main physical model parameters, which coincide with experimental data and the estimates obtained previously by other methods.
PubDate: 2024-07-01
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- Stabilization of the front in a medium with discontinuous characteristics
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Abstract: We study the autowave front propagation in a medium with discontinuous characteristics and the conditions for its stabilization to a stationary solution with a large gradient at the interface between media in the one-dimensional case. The asymptotic method of differential inequalities, based on constructing an asymptotic approximation of the solution, is the main method of study. We develop an algorithm for constructing such an approximation for the solution of the moving front form in a medium with discontinuous characteristics. The application of such an algorithm requires a detailed analysis of the behavior of the solution in neighborhoods of two singular points: the front localization point and the medium discontinuity point. As a result, we obtain a system of equations for the front propagation speed; this distinguishes this paper from the previously published ones. The developed algorithm can be used to describe autowave propagation in layered media. The results can also be extended to the multidimensional case.
PubDate: 2024-07-01
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- $$n$$ -valued quandles and associated bialgebras
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Abstract: We study \(n\) -valued quandles and \(n\) -corack bialgebras. These structures are closely related to topological field theories in dimensions \(2\) and \(3\) , to the set-theoretic Yang–Baxter equation, and to the \(n\) -valued groups, which have attracted considerable attention or researchers. We elaborate the basic methods of this theory, find an analogue of the so-called coset construction known in the theory of \(n\) -valued groups, and construct \(n\) -valued quandles using \(n\) -multiquandles. In contrast to the case of \(n\) -valued groups, this construction turns out to be quite rich in algebraic and topological applications. We study the properties of \(n\) -corack bialgebras, which play a role similar to that of bialgebras in group theory.
PubDate: 2024-07-01
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- Adiabatic perturbation theory for the vector nonlinear Schrödinger
equation with nonvanishing boundary conditions-
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Abstract: We consider a defocusing Manakov system (vector nonlinear Schrödinger (NLS) system) with nonvanishing boundary conditions and use the inverse scattering transform formalism. Integrable models provide a very useful proving ground for testing new analytic and numerical approaches to studying the vector NLS system. We develop a perturbation theory for the integrable vector NLS model. Evidently, small disturbance of the integrability condition can be considered a perturbation of the integrable model. Our formalism is based on the Riemann–Hilbert problem associated with the vector NLS model with nonvanishing boundary conditions. We use the RH and adiabatic perturbation theory to analyze the dynamics of dark–dark and dark–bright solitons in the presence of a perturbation with nonvanishing boundary conditions.
PubDate: 2024-07-01
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- Triple equivalence of the oscillatory behavior for scalar delay
differential equations-
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Abstract: We study the oscillation of a first-order delay equation with negative feedback at the critical threshold \(1/e\) . We apply a novel center manifold method, proving that the oscillation of the delay equation is equivalent to the oscillation of a \(2\) -dimensional system of ordinary differential equations (ODEs) on the center manifold. It is well known that the delay equation oscillation is equivalent to the oscillation of a certain second-order ODE, and we furthermore show that the center manifold system is asymptotically equivalent to this same second-order ODE. In addition, the center manifold method has the advantage of being applicable to the case where the parameters oscillate around the critical value \(1/e\) , thereby extending and refining previous results in this case.
PubDate: 2024-07-01
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- Boundary control problem for the reaction– advection– diffusion
equation with a modulus discontinuity of advection-
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Abstract: We consider a periodic problem for a singularly perturbed parabolic reaction–diffusion–advection equation of the Burgers type with the modulus advection; it has a solution in the form of a moving front. We formulate conditions for the existence of such a solution and construct its asymptotic approximation. We pose a control problem where the required front propagation law is implemented by a specially chosen boundary condition. We construct an asymptotic solution of the boundary control problem. Using the asymptotic method of differential inequalities, we estimate the accuracy of the solution of the control problem. We propose an original numerical algorithm for solving singularly perturbed problems involving the modulus advection.
PubDate: 2024-07-01
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- Nonlinearity in the inverse problems of orbital dynamics using the example
of potentially hazardous asteroids and outer satellites of Jupiter-
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Abstract: We present the results of a study of nonlinearity in inverse problems of the orbital dynamics of Jupiter’s outer satellites, discovered in 2018–2022, and of potentially hazardous asteroids. The results show that for a more accurate study of orbital uncertainty, we must first find the minimum value of a nonlinearity indicator by varying the initial epoch within the measurable interval for different parametric spaces.
PubDate: 2024-07-01
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- Finite-gap solutions of the real modified Korteweg–de Vries equation
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Abstract: We consider methods for constructing finite-gap solutions of the real classical modified Korteweg–de Vries equation and elliptic finite-gap potentials of the Dirac operator. The Miura transformation is used in both methods to relate solutions of the Korteweg–de Vries and modified Korteweg–de Vries equations. We present examples.
PubDate: 2024-07-01
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