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# Vol. 15, no. 1, 2019

 Zemlyanukhin A. I.,  Bochkarev A. V. Abstract The FitzHugh – Rinzel model is considered, which differs from the famous FitzHugh – Nagumo model by the presence of an additional superslow dependent variable. Analytical properties of this model are studied. The original system of equations is transformed into a third-order nonlinear ordinary differential equation. It is shown that, in the general case, the equation does not pass the Painlevé test, and the general solution cannot be represented by Laurent series. Using the singular manifold method in terms of the Schwarzian derivative, an exact particular solution in the form of a kink is constructed, and restrictions on the coefficients of the equation necessary for the existence of such a solution are revealed. An asymptotic solution is obtained that shows good agreement with the numerical one. This solution can be used to verify the results in a numerical study of the FitzHugh – Rinzel model. Keywords: neuron, FitzHugh – Rinzel model, singular manifold, exact solution, asymptotic solution Citation: Zemlyanukhin A. I.,  Bochkarev A. V., Analytical Properties and Solutions of the FitzHugh – Rinzel Model, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 1, pp. 3-12 DOI:10.20537/nd190101
 Kudryashov N. A. Abstract The integrability of the FitzHugh – Rinzel model is considered. This model is an example of the system of equations having the expansion of the general solution in the Puiseux series with three arbitrary constants. It is shown that the FitzHugh – Rinzel model is not integrable in the general case, but there are two formal first integrals of the system of equations for its description. Exact solutions of the FitzHugh – Rinzel system of equations are given. Keywords: FitzHugh – Rinzel model, Painlevé test, first integral, general solution, exact solution Citation: Kudryashov N. A., On Integrability of the FitzHugh – Rinzel Model, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 1, pp. 13-19 DOI:10.20537/nd190102
 Gumerov A. M.,  Ekomasov E. G.,  Kudryavtsev R. V.,  Fakhretdinov M. I. Abstract The generation and evolution of localized waves on an impurity in the scattering of a kink of the sine-Gordon equation are studied. It is shown that the problem can be considered as excitation of oscillations of a harmonic oscillator by a short external impulse. The external impulse is modeled by the scattering of a kink on an impurity. The influence of the modes of motion of a kink on the excitation energy of localized waves is numerically and analytically studied. The method of collective coordinate for the analytical study is used. The value of this energy is determined by the ratio of the impurity parameters and the initial kink velocity. It is shown that the dependence of the energy (and amplitude) of the generated localized waves on the initial kink velocity has only one maximum. This behavior is observed for the cases of point and extended impurities. Analytical expression for the amplitude of the localized wave in the case of point impurity is obtained. This allows controlling the excitation energy of localized waves using the initial kink velocity and impurity parameters. The study of the evolution of localized impurities under the action of an external force and damping has shown a good agreement with the nondissipative case. It is shown that small values of the external force have no significant effect on the oscillations of localized waves. An analytical expression for the logarithmic decrement of damping is obtained. This study may help to control the parameters of the excited waves in real physical systems. Keywords: sine-Gordon equation, impurity, kink, wave generation Citation: Gumerov A. M.,  Ekomasov E. G.,  Kudryavtsev R. V.,  Fakhretdinov M. I., Excitation of Large-Amplitude Localized Nonlinear Waves by the Interaction of Kinks of the Sine-Gordon Equation with Attracting Impurity, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 1, pp. 21-34 DOI:10.20537/nd190103
 Ingel L. K. Abstract The known nonlinear integral model of a turbulent thermal is generalized to the case of the presence of the horizontal component of its motion relative to the surrounding medium (for example, the floating-up of a thermal in a shear flow). In addition, the possibility of the presence of a heat (buoyancy) source in a thermal is considered. In comparison with the author’s previous work, a solution is investigated for the case of unstable background stratification of the medium. The problem is solved in terms of quadratures. The asymptotics of the solution at large time intervals is analyzed. The solution describes, in particular, the nonlinear effect of the interaction of the horizontal and vertical components of the thermal motion, since each of the components affects the rate of entrainment of the surrounding medium, i. e., the growth rate of the thermal size and, hence, its mobility. Keywords: convection, thermals, turbulence, integral models, shear flows, unstable stratification, nonlinear dynamics, analytical solutions Citation: Ingel L. K., On the Nonlinear Dynamics of Turbulent Thermals in the Shear Flow, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 1, pp. 35-39 DOI:10.20537/nd190104
 Vetchanin  E. V. Abstract The motion of a circular cylinder in a fluid in the presence of circulation and external periodic force and torque is studied. It is shown that for a suitable choice of the frequency of external action for motion in an ideal fluid the translational velocity components of the body undergo oscillations with increasing amplitude due to resonance. During motion in a viscous fluid no resonance arises. Explicit integration of the equations of motion has shown that the unbounded propulsion of the body in a viscous fluid is impossible in the absence of external torque. In the general case, the solution of the equations is represented in the form of a multiple series. Keywords: rigid body dynamics, ideal fluid, viscous fluid, propulsion in a fluid, resonance Citation: Vetchanin  E. V., The Motion of a Balanced Circular Cylinder in an Ideal Fluid Under the Action of External Periodic Force and Torque, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 1, pp. 41-57 DOI:10.20537/nd190105
 Ryabov P. E.,  Sokolov S. V. Abstract A completely Liouville integrable Hamiltonian system with two degrees of freedom describing the dynamics of two vortex filaments in a Bose – Einstein condensate enclosed in a cylindrical trap is considered. For the system of two vortices with identical intensities a bifurcation of three Liouville tori into one is detected. Such a bifurcation is found in the integrable case of Goryachev – Chaplygin – Sretensky in rigid body dynamics. Keywords: Vortex dynamics, Bose – Einstein condensate, completely integrable Hamiltonian systems, bifurcation diagram of momentum mapping, bifurcations of Liouville tori Citation: Ryabov P. E.,  Sokolov S. V., Phase Topology of Two Vortices of Identical Intensities in a Bose – Einstein Condensate, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 1, pp. 59-66 DOI:10.20537/nd190106
 Salatich A. A.,  Slavyanov S. Y. Abstract Different forms of the double confluent Heun equation are studied. A generalized Riemann scheme for these forms is given. An equivalent first-order system is introduced. This system can be regarded from the viewpoint of the monodromy property. A corresponding Painlevé equation is derived by means of the antiquantization procedure. It turns out to be a particular case of $P^3$. A general expression for any Painlevé equation is predicted. A particular case of the Teukolsky equation in the theory of black holes is examined. This case is related to the boundary between spherical and thyroidal geometries of a black hole. Difficulties for its antiquantization are shown. Keywords: Double confluent Heun equation, antiquantization, Painlevé equation $P^3$, Teukolsky equation Citation: Salatich A. A.,  Slavyanov S. Y., Antiquantization of the Double Confluent Heun Equation. The Teukolsky Equation, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 1, pp. 79-85 DOI:10.20537/nd190108
 Pranevich A. F. Abstract We consider Hamiltonian systems with $n$ degrees of freedom. Among the general methods of integration of Hamiltonian systems, the Poisson method is of particular importance. It allows one to find the additional (third) first integral of the Hamiltonian system by two known first integrals of the Hamiltonian system. In this paper, the Poisson method of building first integrals of Hamiltonian systems by integral manifolds and partial integrals is developed. Also, the generalization of the Poisson method for general ordinary differential systems is obtained. Keywords: Hamiltonian system, Poisson’s theorem, first integral, integral manifold, partial integral, Poisson bracket Citation: Pranevich A. F., On Poisson’s Theorem of Building First Integrals for Ordinary Differential Systems, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 1, pp. 87-96 DOI:10.20537/nd190109