Theoretical and Mathematical Physics最新文献

We consider various pentagon identities realized by hyperbolic hypergeometric functions and investigate their degenerations to the level of complex hypergeometric functions. In particular, we show that one of the degenerations yields the complex binomial theorem, which coincides with the Fourier transformation of the complex Euler beta integral evaluation. At the bottom, we obtain a Fourier transformation formula for the complex gamma function. This is done with the help of a new type of the limit (omega_1+omega_2to 0) (or (bto i) in two-dimensional conformal field theory) applied to hyperbolic hypergeometric integrals.

We investigate the properties of the Pauli–Jordan–Dirac anticommutator of the quantum field theory of free Dirac electrons in a discrete representation in Minkowski space. We present a method for estimating the number of zeros of the anticommutator on spatial intervals.

A transformation of the Lax representation of chiral-type systems is obtained. Under some additional conditions, this transformation leads to the Lax representation of a new chiral-type system.

The Goeritz matrix is an alternative to the Kauffman bracket and, in addition, makes it possible to calculate the Jones polynomial faster with some minimal choice of a checkerboard surface of a link diagram. We introduce a modification of the Goeritz method that generalizes the Goeritz matrix for computing the HOMFLY–PT polynomials for any (N) in the special case of bipartite links. Our method reduces to purely algebraic operations on matrices, and therefore, it can easily be implemented as a computer program. Bipartite links form a rather large family, including a special class of Montesinos links constructed from the so-called rational tangles. We demonstrate how to obtain a bipartite diagram of such links and calculate the corresponding HOMFLY–PT polynomials using our developed generalized Goeritz method.

We propose a mathematical model describing the transport of sand grains driven by wind, coupled with the evolution of the air momentum, within a bounded domain of (mathbb{R}^2). The sand grains are assumed to be spherical, of constant mass, and are modeled at a mesoscopic scale by a particle density function, while the wind is described at a macroscopic scale by a mean velocity field. We focus on the stationary regime of the system, corresponding to an equilibrium state under constant conditions. The analysis consists of solving the transport equation along characteristics for a given velocity field, followed by the study of the stationary wind equation using Schauder’s fixed-point theorem.

We address the question of establishing a relationship between kinetic equations and continuous medium equations based on group analysis methods for differential equations. Continuing our previous studies, we consider the simplest one-dimensional case. We examine equations with a submaximal (three-dimensional) symmetry group. It turns out that the presence of an external force field requires a radical revision of the problem setting, not only in terms of the use of group methods, but also in the general context. Namely, it turns out that it is impossible to consider the continuous medium equations as a reduced system of moment equations, since, generally speaking, there are merely no high-order moments (in our case, of orders greater than one). However, the use of group methods allows us to overcome this crisis situation (from the point of view of widely held beliefs) by passing to an actually equivalent setting of the problem, which affects neither the moment system nor the nonexisting moment variables. The problem setting transformed in this way turned out, on the one hand, to be related to the well-known group stratification method, somewhat expanding the very understanding of this method, and, on the other hand, providing a simple and effective general method for solving the problem of the relationship between kinetic equations and continuum medium equations.

We show that the negative-order Korteweg–de Vries equation can be integrated using the inverse spectral problem method. We find the evolution of the spectral data of the Sturm–Liouville operator with a periodic potential associated with the finite-gap solution of the negative-order Korteweg–de Vries equation. The obtained results allow the inverse problem method to be applied to solve the negative-order Korteweg–de Vries equation in the class of periodic functions. We prove important implications regarding the analyticity and the spatial period of the finite-gap solution. We show that the solution constructed by the Dubrovin system of equations and the first trace formula satisfies the negative-order Korteweg–de Vries equation.

We investigate the magnetodynamic oscillations of a bounded annular cylinder filled with a highly dense liquid (tar) under the influence of a spatially varying magnetic field. The governing eigenvalue relation is derived, rigorously analyzed, and interpreted within a physical framework. Several well-established stability criteria from different models emerge as limiting cases and are systematically examined. The analysis covers both axisymmetric and nonaxisymmetric perturbation modes, offering a comprehensive understanding of the system’s stability characteristics. The results indicate that the magnetic field exerts a stabilizing effect, while surface tension plays a dual role—suppressing short-wavelength perturbations but amplifying long-wavelength ones. This research has broad implications across engineering and industrial domains, including optimizing electrical generator performance, advancing plasma control techniques, and improving the understanding of magnetically influenced fluid dynamics. Moreover, its findings extend to diverse applications, from aerodynamics (such as airflow over wings) to pipeline design, offering new insights into complex magnetic systems within applied physics, electrical engineering, and magnetism.

We establish a rigorous Riemann–Hilbert (RH) framework for the coupled modified Yajima–Oikawa system. Unlike conventional approaches that initiate spectral analysis with the spatial Lax operator, we employ its temporal counterpart to derive the requisite analytic spectral functions and thereby formulate the associated RH problem. We then obtain explicit multi-soliton solutions in the reflectionless case. Leveraging symbolic computations in Maple, we analyze the ensuing soliton dynamics and illustrate their interactions. Our RH methodology not only elucidates the intricate spectral features of the coupled modified Yajima–Oikawa system but also provides a systematic procedure for constructing its general (N)-soliton solutions.

We investigate several kinds of localized wave solutions of the ((2+1))-dimensional generalized nonlocal Mel’nikov system by using the bilinear method and long-wave limit technique, which have found extensive applications in nonlinear wave theory, optics, and dynamical systems. We derive (N)-soliton solutions by applying Hirota’s bilinear method and obtain breather solutions as well as breathers under periodic wave backgrounds through appropriate parameter restrictions. Taking the long-wave limit of soliton solutions yields the rational solutions, whereby kink-shaped rogue waves and lump solutions under a constant background are constructed. In particular, the dynamical characteristics of kink-shaped rogue waves are analyzed via asymptotic methods. Additionally, semi-rational solutions are obtained with the aid of the partial long-wave limit method, which includes (1) rogue waves and lumps under periodic wave backgrounds, and (2) interaction solutions between breathers and lumps. These analytical approaches enable multidimensional studies of nonlinear wave structures, providing a theoretical foundation for predicting novel wave behaviors in nonlocal nonlinear systems.