Theoretical and Mathematical Physics最新文献

A stationary AKNS hierarchy is investigated through a new method for constructing integrable symplectic mappings. To achieve this, semi-discrete equations are derived from the AKNS spectral problem. Based on these equations, auto-transformations of the stationary AKNS equations are established. These auto-transformations are then used to generate integrable symplectic mappings. Additionally, group properties of these mappings are analyzed and presented.

In this paper, we consider the stochastic fifth-order KdV equation, along with its Lax pair, under the influence of Gaussian white noise and Brownian motion. One new result in this paper is that the soliton-periodic mixed solution can be viewed as a novel tool for generating rogue waves when the soliton solution is in the dominant position. By applying the classical Darboux transformation, we obtain analytic solutions to this equation in determinant form. Through detailed analysis of spectral parameters, we construct soliton solutions, periodic solutions, and their mixed solutions for the stochastic fifth-order KdV equation, which incorporates noise terms. We also consider the generalized Darboux transformation and obtain rational solutions to the stochastic fifth-order KdV equation.

The Riccati equation method and other methods are used to establish global solvability, stability, and oscillation criteria for a class of two–dimensional nonlinear systems of ordinary differential equations. The class under study is a generalization of wide classes of second-order nonlinear ordinary differential equations, studied by many authors. The applicability of some of these criteria is illustrated by examples.

We formulate and investigate a mathematical model of the (mathbf{S^{(0)}}) cosmological system based on a classical scalar field with self-interaction and an ideal scalar-neutral fluid. For a nonzero fluid energy density, the system of equations for the perturbations is reduced to a Lifshitz–Khalatnikov form, which is used to study the cosmological evolution of perturbations at singular points of the (mathbf{S^{(0)}}) background model. At these points, the scalar field, on the one hand, and the gravitational perturbations and the fluid, on the other, become independent subsystems. We find exact solutions for the perturbations of the scalar field, gravitational perturbations, and the perturbation of the energy-momentum of the fluid for its nonrelativistic and ultrarelativistic states. Near stable singular points of the background, the perturbations decay, while near unstable singular points, they grow exponentially rapidly. We find an asymptotic solution to the equation for the perturbation of a scalar field for sufficiently large wavenumbers. This solution is used to establish necessary and sufficient conditions for system instability and evolution. We establish laws for scaling the results of perturbation theory to the parameters of known field-theoretical interaction models.

We study multidimensional integral equations with monotonic and odd nonlinearity. These equations have applications in the dynamic theory of (p)-adic strings. In particular, when the nonlinearity is power-law and the kernels are represented as Gaussian distributions, these equations describe the dynamics (rolling) of (p)-adic open or open–closed strings for a scalar tachyon field. Under certain restrictions on nonlinearity and kernels, we prove the constructive solvability of the equation in the space of continuous and bounded functions. We establish the uniform convergence of the corresponding successive approximations (with a rate of infinitely decreasing geometric progression) to the solution. We prove that the equation under study can simultaneously have alternating and sign-preserving bounded solutions. The results obtained are applied to specific problems in the dynamic theory of (p)-adic strings and in solving a nonlinear boundary value problem for the heat conduction equation.

We consider the Cauchy problem for the two-dimensional massless Dirac equation in graphene with a constant electric field. It is assumed that at the initial time, a localized wave function describes quasi-electrons with momenta lying in the right half-plane. We describe the effect based on the phenomenon of changing the multiplicity of terms (characteristics), which leads to Klein tunneling and consists in the fact that, after some time, a hole component appears in addition to the wave function for the electron component. The components move in opposite directions, and the hole component localizes near a moving point.

In this paper we focus on the application of the (bar{partial})-dressing method to the three-component coupled time-varying coefficient complex mKdV equation. Based upon a ((4 times 4))-matrix (bar{partial})-problem and two linear equations of the spectral transformation matrix, we derive the Lax pair and infinitely many conservation laws for the three-component coupled time-varying coefficient complex mKdV equation. Besides, we construct a hierarchy of the three-component coupled time-varying coefficient complex mKdV equation with a source term by making use of the recursion operator. We derive symmetry conditions of the spectral transformation matrix. We establish (N)-solution solutions and multi-pole solutions for the three-component coupled time-varying coefficient complex mKdV equation and express them in compact forms based on an explicit spectral transformation matrix.

We obtain an explicit solution of the Riemann–Hilbert problem on an elliptic curve for the two-dimensional commutative monodromy representations. By an arbitrary set of points together with a representation of the fundamental group of the curve punctured at these points, we construct a semistable holomorphic vector bundle of degree zero with a logarithmic connection possessing the required singularities and monodromy.

We present a description of the classical elliptic ({rm BC}_1) Ruijsenaars–van Diejen model with eight independent coupling constants through a pair of ({rm BC}_1) type classical Sklyanin algebras generated by the (classical) quadratic reflection equation with non-dynamical (XYZ) (r)-matrix. For this purpose, we consider the classical version of the (L)-operator for the Ruijsenaars–van Diejen model proposed by O. Chalykh. In the ({rm BC}_1) case, it is factorized into the product of two Lax matrices depending on four constants. Then we apply an IRF-Vertex type gauge transformation and obtain a product of the Lax matrices for the Zhukovsky–Volterra gyrostats. Each of them is described by the ({rm BC}_1) version of the classical Sklyanin algebra. In particular case, when four pairs of constants coincide, the ({rm BC}_1) Ruijsenaars–van Diejen model exactly coincides with the relativistic Zhukovsky–Volterra gyrostat. Explicit change of variables is obtained. We also consider another special case of the ({rm BC}_1) Ruijsenaars–van Diejen model with seven independent constants. We show that it can be reproduced by considering the transfer matrix of the classical (1)-site (XYZ) chain with boundaries. In the end of the paper, using another gauge transformation, we represent Chalykh’s Lax matrix in a form depending on Sklyanin’s generators.

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.