Theoretical and Mathematical Physics最新文献

The Riemann–Hilbert approach is applied to the mixed Chen–Lee–Liu equation. The corresponding Jost solutions are found, the analytic, asymptotic and symmetric properties of Jost solutions are studied, and a modified Riemann–Hilbert problem is constructed that satisfies the normalization condition. The formulas for multiple solitons related to the simple poles of the Riemann–Hilbert problem are given in determinant form. According to the Cauchy–Binet formula, the formulas for multiple solitons are given explicitly. Based on these explicit formulas, the first- and second-order solitons are obtained, and the multiple-soliton collisions are verified to be elastic.

We investigate the properties of the Pauli–Jordan–Dirac anticommutator of the quantum field theory of free Dirac electrons in a discrete representation in the spatially one-dimensional case and present a method for estimating the number of zeros of the anticommutator on spatial intervals using the Kronecker theorem.

Using the spectral transform on a torus, we solve the initial–boundary value problem for quasi-one-dimensional excitations in a semibounded ferromagnet, taking the exchange interaction, orthorhombic anisotropy, and magnetostatic fields into account. We used the mixed boundary conditions whose limit cases correspond to free and fully pinned spins at the sample edge. We predict and analyze new types of solitons (moving domain walls and precessing breathers), whose cores are strongly deformed near the sample boundary. At large distances from the sample surface, they take the form of typical solitons in an unbounded medium. We analyze the properties of the reflection of solitons from the sample boundary depending on the degree of spin pinning at the surface. We obtain new conservation laws that guarantee the true boundary conditions to hold when solitons reflect from the sample surface.

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.

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.

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.

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.

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.

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.

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.