Theoretical and Mathematical Physics最新文献

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.

We study the local structure of vector fields on (mathbb{R}^3) that preserve the Martinet (1)-form (alpha=(1+x)dypm z,dz). We classify their singularities up to diffeomorphisms that preserve the form (alpha), as well as their transverse unfoldings. We are thus able to provide a fairly complete list of the bifurcations such vector fields undergo.

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.

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.

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.

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.

We analyze the asymptotic behavior of the Hankel determinant generated by a semiclassical Laguerre weight. For this, we use ladder operators and track the evolution of parameters to establish that an auxiliary quantity associated with the semiclassical Laguerre weight satisfies the Painlevé IV equation, subject to suitable transformations of variables. Using the Coulomb fluid method, we derive the large-(n) expansion of the logarithmic form of the Hankel determinant. This allows us to gain insights into the scaling and fluctuations of the determinant, providing a deeper understanding of its behavior in the semiclassical Laguerre ensemble. Moreover, we delve into the asymptotic evaluation of monic orthogonal polynomials with respect to the semiclassical Laguerre weight, focusing on a special case. In doing so, we shed light on the properties and characteristics of these polynomials in the context of the ensemble. Furthermore, we explore the relation between the second-order differential equations satisfied by the monic orthogonal polynomials with respect to the semiclassical Laguerre weight and the tri-confluent Heun equations or the bi-confluent Heun equations.

We introduce a new equation (a class of equations) to be considered as a candidate for the equation for a nonzero-mass neutrino.

We show how the well-known large-mass expansion of Feynman integrals can be simplified to obtain more terms of the expansion in analytic form. Expansion of two-loop four-point Feynman integrals that contribute to the (H to ggg) process is used as an example.