Georgy Alfimov

    Pas. 4806, bld.5, Zelenograd, Moscow, Russia, 124498
    Moscow Institute of Electronic Technology


    Alfimov G. L., Lebedev M. E.
    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 one-to-one correspondence with bi-infinite 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 map-over-period (monodromy operator) for this equation. Finally, we describe an algorithm that allows one to sketch plots of solutions by its codes.
    Keywords: Duffing-type equation, periodic coefficients, symbolic dynamics, Smale horseshoe
    Citation: Alfimov G. L., Lebedev M. E.,  Complete Description of Bounded Solutions for a Duffing-Type Equation with a Periodic Piecewise Constant Coefficient, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 4, pp.  473-506
    Alfimov G. L.
    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 (so-called E-type kernel). In the simplest particular case, (so-called Kac—Baker kernel) it is shown that the solutions of this equation form a 3-parametric 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 E-type kernel the 3-parametric 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.
    Keywords: nonlocal nonlinear wave equation
    Citation: Alfimov G. L.,  On the dimension of the set of solutions for nonlocal nonlinear wave equation, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 2, pp.  209-226
    Alfimov G. L.
    Nonlocal sine-Gordon equation arises in numerous problems of modern mathematical physics, for instance, in Josephson junction models and lattice models with long-range 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 sine-Gordon 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.
    Keywords: sine-Gordon equation, nonlocality, separatrices
    Citation: Alfimov G. L.,  Nonlocal sine-Gordon equation: kink solutions in the weak nonlocality limit, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 4, pp.  585-602
    Alfimov G. L., Zezyulin D. A.
    The paper concerns non-one-dimensional 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.
    Keywords: Bose–Einstein condensate, solitons, Gross–Pitaevskii equation
    Citation: Alfimov G. L., Zezyulin D. A.,  Demonstrative computation of vortex structures in Bose–Einstein Condensate, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 2, pp.  215-235

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