Georgy Alfimov
Pas. 4806, bld.5, Zelenograd, Moscow, Russia, 124498
Moscow Institute of Electronic Technology
Publications:
Alfimov G. L., Lebedev M. E.
Complete Description of Bounded Solutions for a DuffingType Equation with a Periodic Piecewise Constant Coefficient
2023, Vol. 19, no. 4, pp. 473506
Abstract
We consider the equation $u_{xx}^{}u+W(x)u^3=0$ where $W(x)$ is a periodic alternating piecewise constant function. It is proved that under certain conditions for $W(x)$ solutions of this equation, which are bounded on $\mathbb{R}$, $u(x)<\xi$, can be put in onetoone correspondence with biinfinite sequences of numbers $n\in \{N,\,\ldots,\,N\}$ (called ``codes'' of the solutions). The number $N$ depends on the bounding constant $\xi$ and the characteristics of the function $W(x)$. The proof makes use of the fact that, if $W(x)$ changes sign, then a ``great part'' of the solutions are singular, i.e., they tend to infinity at a finite point of the real axis. The nonsingular solutions correspond to a fractal set of initial data for the Cauchy problem in the plane $(u,\,u_x^{})$. They can be described in terms of symbolic dynamics conjugated with the mapoverperiod (monodromy operator) for this equation. Finally, we describe an algorithm that allows one to sketch plots of solutions by its codes.

Alfimov G. L.
On the dimension of the set of solutions for nonlocal nonlinear wave equation
2011, Vol. 7, No. 2, pp. 209226
Abstract
Nonlocal generalizations of nonlinear wave equation arise in numerous physical applications. It is known that switching from local to nonlocal description may result in new features of the problem and new types of solutions. In this paper the author analyses the dimension of the set of travelling wave solutions for а nonlocal nonlinear wave equation. The nonlocality is represented by the convolution operator which replaces the second derivative in the dispersion term. The results have been obtained for the case where the nonlinearity is bounded, and the kernel of the convolution operator is represented by a sum of exponents with weights (socalled Etype kernel). In the simplest particular case, (socalled Kac—Baker kernel) it is shown that the solutions of this equation form a 3parametric set (assuming the equivalence of the solutions which differ by a shift with respect to the independent variable). Then it is shown that in the case of the general Etype kernel the 3parametric set of solutions also exists, generically, under some additional restrictions. The word «generically» in this case means some transversality condition for intersection of some manifolds in a properly defined phase space.

Alfimov G. L.
Nonlocal sineGordon equation: kink solutions in the weak nonlocality limit
2009, Vol. 5, No. 4, pp. 585602
Abstract
Nonlocal sineGordon equation arises in numerous problems of modern mathematical physics, for instance, in Josephson junction models and lattice models with longrange interactions. Kink solutions of this equation correspond to physically relevant objects such as magnetic flux vortex in Josephson electrodynamics. In this paper the kink solutions for the nonlocal sineGordon equation are considered in weak nonlocality limit. In this limit the equation for travelling waves can be reduced to ordinary differential equation of 4th order with two governing parameters. A survey of possible kink solutions for this equation and for all combination of the governing parameters is presented. The collection of known results is given for the regions on the plane of model parameters which just have been investigated. New results of qualitative and numerical analysis are reported for other regions of the plane of model parameters.

Alfimov G. L., Zezyulin D. A.
Demonstrative computation of vortex structures in Bose–Einstein Condensate
2009, Vol. 5, No. 2, pp. 215235
Abstract
The paper concerns nononedimensional structures described by nonlinear SchrЁodinger equation with additional potential term. A method for numerical construction of structures of such kind is suggested. The method is based on dynamical interpretation of the equation under consideration. Some exact statements are formulated; they allow (in some cases) to perform demonstrative computation and to list all the types of structures mentioned above. Physical applications of the problem are associated with the theory of a Bose—Einstein condensate. In this context the considered equation is called Gross—Pitaevskii equation and the structures under consideration correspond to macroscopic wave function of the condensate.
