Theoretical and Mathematical Physics最新文献

We study the three-loop gauge (beta)-functions in general (mathcal N=1) supersymmetric gauge theories regularized by higher-covariant-derivatives (HCD) supplemented with Pauli–Villars subtraction. The all-structure three-loop form of the (beta)-functions in the HCD framework is known and involves regulator-dependent parameters. Here we evaluate these parameters explicitly for the exponential regulators (R(x)=e^{x^n}) and (F(x)=e^{x^m}). We obtain the constants (A(n)) and (B(m)) in closed form, together with their large-(n,m) asymptotics, and substitute them into the general three-loop expressions. This yields fully explicit, regulator-parameterized (beta)-functions and a systematic expansion in (1/n) and (1/m) that organizes finite, scheme-dependent terms. We then introduce finite coupling redefinitions that map the renormalized (overline{mathrm{DR}}) result to a scheme compatible with the Novikov–Shifman–Vainshtein–Zakharov relation. Our analysis clarifies how exponential higher-derivative regulators preserve this relation at the bare level and illustrates the regulator-driven structure of supersymmetric renormalization group flows.

In this paper, we provide a partial solution to the Birman–Williams problem on describing the planetary link of a fibered knot (Ksubsetmathbb S^3). Using Nielsen’s classification theory of periodic homeomorphisms of surfaces and its close connection with the theory of gradient-like diffeomorphisms, we prove that the planetary link of the trefoil knot (the unique periodic fibered knot of genus (1)) is the link consisting of the trefoil and the Hopf link. Furthermore, we classify all flows on the 3-sphere (mathbb{S}^3) that realize the planetary link of the trefoil, i.e., structurally stable flows with a minimal number of periodic orbits transverse to the fibration of the trefoil.

We reveal the most general regularities of the surface distribution of compressions and stretches in a surfactant film distributed on a liquid surface, which is in an unstable state with respect to the excess of the surface electric charge.

We classify hyperbolic-type equations and study the class of equations in the form

where (u(x,y)) is the unknown function and (x,y) are independent variables. The classification is based on the requirement for the existence of higher symmetries of the fifth order. As a result, we obtain a list of four equations satisfying the specified conditions. One of these equations is new.

The present work is devoted to a system of fast and slow equations with spatially discontinuous nonlinear advection coefficients and source functions. We investigate a solution with an internal transition layer localized near the discontinuity point. This study continues earlier research on stationary fast-slow systems of the reaction–diffusion–advection type, in which a boundary-layer solution of a similar problem with smooth advection coefficients and reactive terms was considered. Furthermore, this work is a natural extension of an earlier investigation of a similar system with discontinuous reactive terms but without advective terms, and addresses the question of the influence of these advective terms on the structure of the transition layer. The main results include the construction of an asymptotic approximation for the stationary solution, alongside proofs of its existence and asymptotic stability as a solution of the corresponding initial-boundary value problem. The proof is based on the asymptotic method of differential inequalities.

The equation (displaystyleint_{0}^{infty} T(|x - t|)f(t),dt = g(x)), where the functions (T) and (g) satisfy certain smoothness, monotonicity, and integrability conditions, is considered. By applying double differentiation, an exact solution to this equation is constructed in terms of locally integrable functions and the Dirac delta function.

We study the integrability problem of the nonuniform Hirota (vc-Hirota) equation with variable coefficients by using the Bell polynomial method and the Darboux transformation (DT). The nonuniform vc-Hirota equation describes the wave propagation of picosecond pulses more accurately than the nonlinear Schrödinger (NLS) equation. Firstly, by using the Bell polynomial method, we construct the Bäcklund transformation (BT) and Lax pair of the equation to prove that the nonuniform vc-Hirota equation is Lax integrable. Interestingly, the Lax pair we constructed has variable coefficients and is nonisospectral, as opposed to having constant coefficients. Secondly, by pairing the Lax pair with the zero-curvature equation, we obtain the DT of the equation. Finally, we construct the soliton-like solutions of the equation.

The modified Korteweg–de Vries equation of negative order is investigated by the inverse scattering transform method in the case where the corresponding non-self-adjoint Dirac-type operator has a finite number of discrete multiple eigenvalues of a given multiplicity. The evolution laws of the scattering data, including the normalization chains corresponding to the multiple eigenvalues, are derived. A representation of the solution to the Cauchy problem is constructed based on the Gelfand–Levitan–Marchenko integral equations adapted to the degenerate spectral structure.

In this study, a symmetry analysis is performed in order to probe the impact of rivers on the Earth’s mass balance within a new symmetry framework. As a river of the highest discharge, the Amazon River was modeled to probe its impact on continental mass redistribution. It was shown that roughly 1% of South America’s continental mass would be separated from the Upper Amazon Basin within 155 years, progressing sequentially along the river to reach 10% near the Amazon mouth over 1550 years. The resulting asymmetry in mass distribution is illustrated by synthetic two-dimensional hysteresis curves, confirming the river’s role as a localized symmetry-breaking factor.

We investigate a supercritical traveling wave as a solution to a hyperbolic semilinear partial differential equation and establish the conditions for its existence. We focus primarily on the stability problem, which leads us to analyze a spectral problem for a non-self-adjoint operator. Finally, we describe a wide class of initial perturbations with respect to which the wave exhibits asymptotic stability in the linear approximation.