A classical version of the lattice sine-Gordon model

A completely integrable regularized lattice sine-Gordon model is considered by the method of the inverse scattering problem. It is shown that the variables of action-angle type for the lattice and continuous models coincide. Solitons of the lattice model are constructed, and their scattering is described.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 120, pp. 173–187, 1982.     

Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise

We study the asymptotic behavior of solutions to the stochastic sine-Gordon lattice equations with multiplicative white noise. We first prove the existence and uniqueness of solutions, and then establish the existence of tempered random bounded absorbing sets and global random attractors.     

New approach to the solution of the classical sine-Gordon equation and its generalizations

We obtain new exact solutions U(x, y, z, t) of the three-dimensional sine-Gordon equation. The three-dimensional solutions depend on an arbitrary function F(α) whose argument is a function α(x, y, z, t). The ansatz α is found from an equation linear in (x, y, z, t) whose coefficients are arbitrary functions of α that should satisfy a system of algebraic equations. By this method, we solve the classical and a generalized sine-Gordon equation; the latter additionally contains first derivatives with respect to (x, y, z, t). We separately consider an equation that contains only the first derivative with respect to time. We present approaches to the solution of the sine-Gordon equation with variable amplitude. The considered methods for solving the sine-Gordon equation admit a natural generalization to the case of integration of the same types of equations in a space of arbitrarily many dimensions.     

On the absence of radiationless motion for a rotating classical charge

For the Abraham-Lorentz model of a spinning charge a new approach is used to prove that all solutions converge to the set of stationary solutions in the limit t→±∞. This new method allows one to get rid of the additional assumptions that have been imposed before (e.g., the Wiener condition).     

Solutions of the Kirkwood–Salsburg equation for a lattice classical system of one-dimensional oscillators with positive finite-range many-body interaction potentials

For a system of classical one-dimensional oscillators on a d-dimensional hypercubic lattice that interact via a pair superstable potential and many-body positive finite potentials, the (lattice) Kirkwood–Salsburg equation is proposed and solved for the first time. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 10, pp. 1427–1433, October, 2008.     

Solutions of the classical equation of motion for a spin in electromagnetic fields

The evolution of a charged particle spin in electromagnetic fields of a special form is investigated based on the Bargmann-Michel-Telegdi equation. For some types of nonhomogeneous fields, an exact analytic form of the dependence of the spin vector on the four-dimensional velocity of the particle is found. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 3, pp. 509–518, December, 1999.