Mathematical Physics Research Articles

Super fiber bundles, connection forms, and parallel transport

The present work provides a mathematically rigorous account on super fiber bundle theory, connection forms, and their parallel transport, which ties together various approaches. We begin with a detailed introduction to super fiber bundles. We then introduce the concept of so-called relative supermanifolds as well as bundles and connections defined in these categories. Studying these objects turns out to be of utmost importance in order to, among other things, model anticommuting classical fermionic fields in mathematical physics. We then construct the parallel transport map corresponding to such connections and compare the results with those found by other means in the mathematical literature. Finally, applications of these methods to supergravity will be discussed, such as the Cartan geometric formulation of Poincaré supergravity as well as the description of Killing vector fields and Killing spinors of super Riemannian manifolds arising from metric reductive super Cartan geometries.

Some features of oscillations and stability of reinforced cylindrical shells under action of movable inertial loading

This paper describes some features of the mathematical models for the elastic shell with movable load and for the elastic elements of changeable length. In these systems two forms of eigenoscillations - the eigen component and the accompanying one, displaced in phase to the right angle correspond to every frequency of the system. The accompanying component is caused by the mobile inertia load or by the changeable length and they are non-trivial when those factors are present. As well as for objects with time-varying length, these problems lie in outside of the scope of the classical problems of mathematical physics due to that the eigenfrequencies and eigenforms will be presented as time-nondependent functions. This non-classical section of the mathematical physics is waiting for its development, new researches and generalizations. The main attention of this work is paid to the elastic systems with the mobile inertia load. The problem of oscillations and stability of thin cylindrical envelope with regular stringer set is being considered under assumption that they are regularly placed with a small step from one to another. Also, we assume that there is unsymmetrical stream of mobile inertia load. There is a wide variety of problems in area of jet engine, cooling systems, flying, and floating vehicle framework’s design, which boil down to such computational schemes. The linear theory of structural-orthotropical envelopes with equilibrium equations and geometrical Donnel-Mushtary relations are being used to obtain of the discussed problem formulation. We assume that ribs of the shell are considered as Kirchhoff-Klebsh bars. The accurate solution of the problem is obtained by using of two-waves representation for motion and it has been applied for research on influence of geometry, elastic and structural parameters on the oscillation and stability properties of the shell.

IDENTIFICATION OF THE RIGHT HAND SIDE OF A QUASILINEAR PSEUDOPARABOLIC EQUATION WITH MEMORY TERM

The study of equations of mathematical physics, including inverse problems, is relevant today. This work is devoted to the fundamental problem of studying the solvability and qualitative properties of the solution of the inverse  problem  for  a  quasilinear  pseudoparabolic  equation (also called Sobolev-type equations) with memory term. To date, studies of direct and inverse problems for a pseudoparabolic equations are rapidly developing in connection with the needs of modeling and control of processes in thermal physics, hydrodynamics, and mechanics of a continuous medium. The pseudoparabolic equations similar to those considered in this work arise in the description of heat and mass transfer processes, processes of non-Newtonian fluids motion, wave processes, and in many other areas. The main types of the inverse problems are: boundary, retrospective, coefficient and geometric. The boundary and retrospective inverse problems lead to the study of linear problems. In turn, the statements related to the study of coefficient and geometric types bring to the nonlinear problems. Coefficient inverse problems are divided into two main groups: coefficient inverse problems, where the unknown is a function of one or several variables, and finite-dimensional coefficient inverse problems. In this article the existence and uniqueness of a weak and strong solution of the inverse problem in a bounded domain are proved by the Galerkin method. Also we used Sobolev’s embedding theorems, and obtained a priori estimates for the solution. Moreover, we get local and global theorems on the existence of the solution. Key words: Pseudoparabolic equation, inverse problem, existence, uniqueness, local solvability, global solvability, non-local condition.

Harmonic mappings between singular metric spaces

In this paper, we survey the existence, uniqueness and interior regularity of solutions to the Dirichlet problem associated with various energy functionals in the setting of mappings between singular metric spaces. Based on known ideas and techniques, we separate the necessary analytical assumptions to axiomatizing the theory in the singular setting. More precisely, (1) we extend the existence result of Guo and Wenger (Comm Anal Geom 28(1):89–112, 2020) for solutions to the Dirichlet problem of Korevaar–Schoen energy functional to more general energy functionals in purely singular setting. (2) When Y has non-positive curvature in the sense of Alexandrov (NPC), we show that the ideas of Jost (Calc Var Partial Differ Equ 5(1):1–19, 1997) and Lin (Analysis on singular spaces, collection of papers on geometry, analysis and mathematical physics, World Science Publishers, River Edge, pp 114–126, 1997) can be adapted to the purely singular setting to yield local Hölder continuity of solutions of the Dirichlet problem of Korevaar–Schoen and Kuwae–Shioya. (3) We extend the Liouville theorem of Sturm (J Reine Angew Math 456:173–196, 1994) for harmonic functions to harmonic mappings between singular metric spaces. (4) We extend the theorem of Mayer (Comm Anal Geom 6:199–253, 1998) on the existence of the harmonic mapping flow and solve the corresponding initial boundary value problem. Combing these known ideas, with the more or less standard techniques from analysis on metric spaces based on upper gradients, leads to new results when we consider harmonic mappings from \({{\,\mathrm{RCD}\,}}(K,N)\) spaces into NPC spaces. Similar results for the Dirichlet problem associated with the Kuwae–Shioya energy functional and the upper gradient functional are also derived.

On the Non-Smooth Solutions of 3D Navier-Stokes Equations for the Incompressible Fluid Flows

In the paper non-stationary 3D incompressible viscous fluid flow over the point, the infinite line, the plane, the rectangular prism and the octahedron are studied. The corresponding Navier-Stokes equations (NSE) with the appropriate initial-boundary conditions are considered. NSE is a very important equation and has various applications in Plasma Physics, Astrophysics, magma physics, geophysical fluids, biophysics, nanofluids, etc. NSE describes significant characteristics of different fluids. The exact solutions are obtained in a very few cases and especially in 2D. In the paper the novel exact non-smooth solutions blow-up in time are obtained for the specific pressure and initial conditions by means of the methods of mathematical physics (the main result). Besides, the solutions for the turbulent flows are given. Those solutions are new and are applied to solving of the problem of some substance transportation in the space by means of the turbulent flow. The profiles of the velocity and substance distribution are constructed by means of “Maple” for the different parameters. The results have applications to the description of atmospheric and ocean currents, nanosciences.

Stability analysis and soliton solutions for the longitudinal wave equation in magneto electro-elastic circular rod

This paper presented exact solutions and stability analysis to the nonlinear longitudinal wave equation (NLLWE) in a magneto electro-elastic (MEE) circular rod. The new generalized exponential rational function method (GERFM) has been applied for finding the exact solutions of NLLWE in MEE circular rod. It has the capacity to generate many types of solutions in different form. Therefore, it is effective in extracting new types of solutions in other nonlinear partial differential equations (NLPDEs) that commonly model applications in engineering, mathematical physics. Furthermore, some 2D, 3D graphs are obtained to understand the physical phenomena of this longitudinal wave equation and the stability of governing equation is investigated.

On Fourier–Bessel matrix transforms and applications

The Fourier–Bessel transform is an integral transform and is also known as the Hankel transform. This transform is a very important tool in solving many problems in mathematical sciences, physics, and engineering. Very recently, Abdalla (AIMS Mathematics 6: [2021], 6122–6139) introduced certain Hankel integral transforms associated with functions involving generalized Bessel matrix polynomials and various applications. Motivated by this work, we introduce the Fourier–Bessel matrix transform (FBMT) containing Bessel matrix function of the first kind as a kernel. The corresponding inversion formula and several illustrative examples of this transform have been presented. A relation between the Laplace transform and the FBMT has been obtained. Furthermore, a convolution of the FBMT is constructed with some properties. In addition, some applications are proposed in the present research. Finally, we outlined the significant links for the preceding outcomes of some particular cases with our results.

Fractionally charged anyons generated by topological path fusion of propagating electron in magnetic flux lattice

Anyon is collective excitation of two dimensional electron gas subjected to strong magnetic field, carrying fractional charges and exotic statistical character beyond fermion and boson. So far, anyons with serial fractional charges only exist in fractional quantum Hall effect. It is still a challenge to find new serial of fractional charges in other physical system and develop an unified mathematical physics theory based on the same root. Here a topological path fusion theory of propagating electrons in magnetic flux lattice is proposed to explore the physical origin of fractional charges based on a generalization of Feynman's path integral theory and Thurston's train track theory. This mathematical physics theory generated the existed serial of fractional charges in fractional quantum Hall effect and predicted new serial of fractional charges. A serial of irrational charges are predicted in one dimensional lattice of magnetic fluxes. Fractionally charged anyons are also generated in two dimensional and three dimensional lattice of train tracks of electric currents, revealing an exact correspondence between knot lattice model and train track model. A new explanation for the modular symmetry of complex Hall conductance and composite fermion in fractional quantum Hall effect is also derived from this topological path fusion theory. Experimental observation of anyon in three dimensions can be realized by constructing three dimensional interlocking magnetic fluxes or mapping magnetic fluxes into forbidden zones in multi-connected space filled by solid state material. A photonic crystal with porous nano-structures is a promising system for detecting fractional charges and paves a new way for topological quantum computation.

Geometric R\xe9nyi Divergence and its Applications in Quantum Channel Capacities

Having a distance measure between quantum states satisfying the right properties is of fundamental importance in all areas of quantum information. In this work, we present a systematic study of the geometric Rényi divergence (GRD), also known as the maximal Rényi divergence, from the point of view of quantum information theory. We show that this divergence, together with its extension to channels, has many appealing structural properties, which are not satisfied by other quantum Rényi divergences. For example we prove a chain rule inequality that immediately implies the “amortization collapse” for the geometric Rényi divergence, addressing an open question by Berta et al. [Letters in Mathematical Physics 110:2277–2336, 2020, Equation (55)] in the area of quantum channel discrimination. As applications, we explore various channel capacity problems and construct new channel information measures based on the geometric Rényi divergence, sharpening the previously best-known bounds based on the max-relative entropy while still keeping the new bounds single-letter and efficiently computable. A plethora of examples are investigated and the improvements are evident for almost all cases.

Investigation of Extended k-Hypergeometric Functions and Associated Fractional Integrals

Hypergeometric functions have many applications in various areas of mathematical analysis, probability theory, physics, and engineering. Very recently, Hidan et al. (Math. Probl. Eng., ID 5535962, 2021) introduced the (p, k)-extended hypergeometric functions and their various applications. In this line of research, we present an expansion of the k-Gauss hypergeometric functions and investigate its several properties, including, its convergence properties, derivative formulas, integral representations, contiguous function relations, differential equations, and fractional integral operators. Furthermore, the current results contain several of the familiar special functions as particular cases, and this extension may enrich the theory of special functions.

ON WEIGHTED k-FRACTIONAL OPERATORS WITH APPLICATIONS IN MATHEMATICAL PHYSICS

The main objective of this paper is to present weighted [Formula: see text]-fractional integral and derivative operators of a function with respect to another function and to uncover their properties. In addition to this, we find the weighted Laplace transform of the newly defined operators. As applications of the weighted [Formula: see text]-fractional operators in mathematical physics, we study the fractional forms of kinetic differintegral equation and the time-fractional heat equation involving the novel operators and find their solutions using weighted Laplace transform.

Peridynamics for fluid mechanics and acoustics

Peridynamic governing equations of fluid mechanics for barotropic flow are developed. As a special case of such a flow, linearized acoustics is also considered. The peridynamic governing equation of acoustics is also developed and discussed in detail. In order to obtain these new peridynamic governing equations, integral operators called “peridynamic D operators” are developed systematically, which are obtained by directly requiring the peridynamic D operators to converge to corresponding classical differential operators as the generalized material horizon approaches 0. Even though peridynamic D operators are applied only to fluid mechanics, acoustics, and heat conduction in an anisotropic inhomogeneous material in the present paper, it is clear that these peridynamic D operators can be used in any other field of mathematical physics to obtain a peridynamic (nonlocal) version of the governing equations. As an application of the newly obtained peridynamic governing equations, a time-dependent 3D (three-dimensional) peridynamic acoustics equation with a sound source is analytically solved for two different initial pressure disturbance profiles, and the results are discussed. These are believed to be the first exact analytical solutions for peridynamic acoustics.

On determining some exact wave solutions to the Nizhnik–Novikov–Veselov system via a rebuts technique

In this study, a significant model of mathematical physics used to describe real-world problems, namely the Nizhnik–Novikov–Veselov (NNV) system is studied and various versions of wave solutions are simultaneously explored. The NNV system is considered an essential model to the KdV hierarchy and is widely used to study the scalar nucleons associated with neutral scalar masons. Thus, the exact wave solutions are important to the associated researchers. To this aim, a robust approach named the generalized exponential rational function method (GERFM) is used to achieve this goal. With the aid of GERFM abundant new wave solutions are formally constructed. All generated solutions are constructed with distinct forms that are dependent on free parameters and determined by the self-form of the NNV system. Moreover, the free parameters play the main roles in the dispersive terms and the values of free parameters influence the physical properties of the traveling solitary and periodic waves to the NNV system directly. Finally, some 3D graphs of the generated solutions are provided to show the propagations of traveling waves. The performed results will have great potential applications to optical fibers, magneto-electrodynamics, the collision of heavy ions, and quantum mechanics.

On the modulation instability analysis and deeper properties of the cubic nonlinear Schrödinger’s equation with repulsive [formula omitted]-potential

This projected work applies the generalized exponential rational function method to extract the complex, trigonometric, hyperbolic, dark bright soliton solutions of the cubic nonlinear Schrödinger’s equation. Moreover, trigonometric, complex, strain conditions and dark-bright soliton wave distributions are also reported. Furthermore, the modulation instability analysis is also studied in detail. To better understand the dynamic behavior of some of the obtained solutions, several numerical simulations are presented in the paper. According to the obtained results, it is clear that the method has less limitations than other methods in determining the exact solutions of the equations. Despite the simplicity and ease of use of this method, it has a very powerful performance and is able to introduce a wide range of different types of solutions to such equations. The idea used in this paper is readily applicable to solving other partial differential equations in mathematical physics.

Degenerate Canonical Forms of Ordinary Second-Order Linear Homogeneous Differential Equations

For each fundamental and widely used ordinary second-order linear homogeneous differential equation of mathematical physics, we derive a family of associated differential equations that share the same “degenerate” canonical form. These equations can be solved easily if the original equation is known to possess analytic solutions, otherwise their properties and the properties of their solutions are de facto known as they are comparable to those already deduced for the fundamental equation. We analyze several particular cases of new families related to some of the famous differential equations applied to physical problems, and the degenerate eigenstates of the radial Schrödinger equation for the hydrogen atom in N dimensions.

Extremal statistics for stochastic resetting systems.

While averages and typical fluctuations often play a major role in understanding the behavior of a nonequilibrium system, this nonetheless is not always true. Rare events and large fluctuations are also pivotal when a thorough analysis of the system is being done. In this context, the statistics of extreme fluctuations in contrast to the average plays an important role, as has been discussed in fields ranging from statistical and mathematical physics to climate, finance, and ecology. Herein, we study extreme value statistics (EVS) of stochastic resetting systems, which have recently gained significant interest due to its ubiquitous and enriching applications in physics, chemistry, queuing theory, search processes, and computer science. We present a detailed analysis for the finite and large time statistics of extremals (maximum and arg-maximum, i.e., the time when the maximum is reached) of the spatial displacement in such system. In particular, we derive an exact renewal formula that relates the joint distribution of maximum and arg-maximum of the reset process to the statistical measures of the underlying process. Benchmarking our results for the maximum of a reset trajectory that pertain to the Gumbel class for large sample size, we show that the arg-maximum density attains a uniform distribution independent of the underlying process at a large observation time. This emerges as a manifestation of the renewal property of the resetting mechanism. The results are augmented with a wide spectrum of Markov and non-Markov stochastic processes under resetting, namely, simple diffusion, diffusion with drift, Ornstein-Uhlenbeck process, and random acceleration process in one dimension. Rigorous results are presented for the first two setups, while the latter two are supported with heuristic and numerical analysis.

A new topological perspective of expanding space-times with applications to cosmology

We discuss the possible role of the Tietze extension theorem in providing a rigorous topological base to the expanding space-time in cosmology. A simple toy model has been introduced to show the analogy between the topological extension from a circle [Formula: see text] to the whole space [Formula: see text] and the cosmic expansion from a non-zero volume to the whole space-time in non-singular cosmological models. A topological analogy to the cosmic scale factor function has been suggested, the paper refers to the possible applications of the topological extension in mathematical physics.

Soliton Types Wave Solutions to Fractional Order Nonlinear Evolution Equations Arise in Mathematical Physics

Soliton Types Wave Solutions to Fractional Order Nonlinear Evolution Equations Arise in Mathematical Physics

Lie Symmetry Analysis and Dynamics of Exact Solutions of the (2+1)-Dimensional Nonlinear Sharma–Tasso–Olver Equation

In soliton theory, the dynamics of solitary wave solutions may play a crucial role in the fields of mathematical physics, plasma physics, biology, fluid dynamics, nonlinear optics, condensed matter physics, and many others. The main concern of this present article is to obtain symmetry reductions and some new explicit exact solutions of the (2 + 1)-dimensional Sharma–Tasso–Olver (STO) equation by using the Lie symmetry analysis method. The infinitesimals for the STO equation were achieved under the invariance criteria of Lie groups. Then, the two stages of symmetry reductions of the governing equation are obtained with the help of an optimal system. Meanwhile, this Lie symmetry method will reduce the STO equation into new partial differential equations (PDEs) which contain a lesser number of independent variables. Based on numerical simulation, the dynamical characteristics of the solitary wave solutions illustrate multiple-front wave profiles, solitary wave solutions, kink wave solitons, oscillating periodic solitons, and annihilation of parabolic wave structures via 3D plots.

FUNCTIONAL MODEL FOR BOUNDARY‐VALUE PROBLEMS

We develop a functional model for operators arising in the study of boundary-value problems of materials science and mathematical physics. We then provide explicit formulae for the resolvents of the associated extensions of symmetric operators in terms of the associated generalised Dirichlet-to-Neumann maps, which can be utilised in the analysis of the properties of parameter-dependent problems as well as in the study of their spectra.