Steady soliton-like solutions of Euler's equations with intrinsic force fields

Steady solutions of a system of Euler's equations when there are intrinsic force interactions of electrical and gravitational type are analysed. Particular attention is devoted to the class of soliton “particle-like” solutions with concentrated charge and mass. Periodic steady solutions for the case when the intrinsic gravitational field has non-zero density at infinity is also considered.     

On Darboux solutions of the Euler's equation

Summary We construct a non-constant Darboux functionf: R R which is a solution of the Euler's functional equationf(x + f(x)) = f(x) for everyx. This function is a counter-example to a statement given in the literature.     

Wandering solutions of Euler's D-2 equation

O. Yu. Shmidt Institute of Physics of the Earth, Academy of Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 3, pp. 70–71, July–September, 1991.     

Statistical interpretation of rheological equations

The author examines one of the forms of elementary micromechanisms of formation of irreversible or high-elastic strains due to bending of the polymer chains with restricted internal rotation. He describes a method for the statistical calculation of the elementary micromechanisms that develop under the influence of temperature fluctuations, stress, stress rate, and mean hydrostatic pressure and together constitute the measured macrostrain. The use of the concept of asymmetric media and the introduction of a coefficient of asymmetry, of the structural mechanical state make possible the refinement of the mathematical model of a polymer material and, in particular, permit not only a phenomenological but also a physical statistical treatment of the know Maxwell, Kelvin and Maxwell-Thomson rheological equations.Mekhanika Polimerov, Vol. 2, No. 2, pp. 197–213, 1966     

Statistical mechanics of nonlinear wave equations

The recurrences of initial states for nonlinear wave equations of the form with odd f are studied. The main result refines and generalizes the investigations of Friedlander and is related to the Poincaré theorem for finite dimensional Hamiltonian systems. Bibliography: 7 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 235, 1900, pp. 287–294.     

Differential equations without solutions

Mathematische Annalen -     

Entire solutions of integrodifference equations

This paper is concerned with the existence of nontrivial entire solutions of integrodifference equations. By constructing proper auxiliary integrodifference equations and functions, we present the existence of nontrivial entire solutions even if the birth function is not monotone, which are different from the well-studied travelling wave solutions. The convergence of entire solutions is also given.     

Periodic solutions of difference equations

Consider the difference equation

(∗)     

Periodic solutions of difference equations

We give an overview of results on the existence of periodic solutions of difference equations that have been obtained in the last two decades. The survey covers both ordinary and Volterra difference systems. Some extensions and generalizations of known result are also presented.     

Transcendental solutions of diophantine equations

Orlov State Pedagogical Institute. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 22, No. 4, pp. 78–79, October–December, 1988.