TY - JOUR A1 - Glebov, Sergei A1 - Kiselev, Oleg A1 - Tarkhanov, Nikolai Nikolaevich T1 - Weakly nonlinear dispersive waves under parametric resonance perturbation N2 - We consider a solution of the nonlinear Klein-Gordon equation perturbed by a parametric driver. The frequency of parametric perturbation varies slowly and passes through a resonant value, which leads to a solution change. We obtain a new connection formula for the asymptotic solution before and after the resonance. Y1 - 2010 UR - http://www3.interscience.wiley.com/cgi-bin/issn?DESCRIPTOR=PRINTISSN&VALUE=0022-2526 U6 - https://doi.org/10.1111/j.1467-9590.2009.00460.x SN - 0022-2526 ER - TY - JOUR A1 - Tarkhanov, Nikolai Nikolaevich T1 - The dirichlet to Neumann operator for elliptic complexes JF - Transactions of the American Mathematical Society N2 - We define the Dirichlet to Neumann operator for an elliptic complex of first order differential operators on a compact Riemannian manifold with boundary. Under reasonable conditions the Betti numbers of the complex prove to be completely determined by the Dirichlet to Neumann operator on the boundary. KW - Elliptic complexes KW - Dirichlet to Neumann operator KW - inverse problems Y1 - 2011 SN - 0002-9947 VL - 363 IS - 12 SP - 6421 EP - 6437 PB - American Mathematical Soc. CY - Providence ER - TY - JOUR A1 - Malass, Ihsane A1 - Tarkhanov, Nikolai Nikolaevich T1 - The de Rham Cohomology through Hilbert Space Methods JF - Journal of Siberian Federal University. Mathematics & physics N2 - We discuss canonical representations of the de Rham cohomology on a compact manifold with boundary. They are obtained by minimising the energy integral in a Hilbert space of differential forms that belong along with the exterior derivative to the domain of the adjoint operator. The corresponding Euler-Lagrange equations reduce to an elliptic boundary value problem on the manifold, which is usually referred to as the Neumann problem after Spencer. KW - De Rham complex KW - cohomology KW - Hodge theory KW - Neumann problem Y1 - 2019 U6 - https://doi.org/10.17516/1997-1397-2019-12-4-455-465 SN - 1997-1397 SN - 2313-6022 VL - 12 IS - 4 SP - 455 EP - 465 PB - Sibirskij Federalʹnyj Universitet CY - Krasnoyarsk ER - TY - JOUR A1 - Stepanenko, Victor A1 - Tarkhanov, Nikolai Nikolaevich T1 - The Cauchy problem for Chaplygin's system N2 - We discuss the Cauchy problem for the so-called Chaplygin system which often appears in gas, aero- and hydrodynamics. This system can be thought of as a nonlinear analogue of the Cauchy-Riemann system in the plane. We pose Cauchy data on a part of the boundary and apply variational approach to construct a solution to this ill-posed problem. The problem actually gives insight to fundamental questions related to instable problems for nonlinear equations. Y1 - 2010 UR - http://www.informaworld.com/openurl?genre=journal&issn=1747-6933 U6 - https://doi.org/10.1080/17476930903394978 SN - 1747-6933 ER - TY - JOUR A1 - Kiselev, Oleg M. A1 - Tarkhanov, Nikolai Nikolaevich T1 - The capture of a particle into resonance at potential hole with dissipative perturbation JF - Chaos, solitons & fractals : applications in science and engineering ; an interdisciplinary journal of nonlinear science N2 - We study the capture of a particle into resonance at a potential hole with dissipative perturbation and external periodic excitation. The measure of resonance solutions is evaluated. We also derive an asymptotic formula for the parameter range of those solutions which are captured into resonance. Y1 - 2014 U6 - https://doi.org/10.1016/j.chaos.2013.11.003 SN - 0960-0779 SN - 1873-2887 VL - 58 SP - 27 EP - 39 PB - Elsevier CY - Oxford ER - TY - JOUR A1 - Mera, Azal A1 - Stepanenko, Vitaly A. A1 - Tarkhanov, Nikolai Nikolaevich T1 - Successive approximation for the inhomogeneous burgers equation JF - Journal of Siberian Federal University : Mathematics & Physics N2 - The inhomogeneous Burgers equation is a simple form of the Navier-Stokes equations. From the analytical point of view, the inhomogeneous form is poorly studied, the complete analytical solution depending closely on the form of the nonhomogeneous term. KW - Navier-Stokes equations KW - classical solution Y1 - 2018 U6 - https://doi.org/10.17516/1997-1397-2018-11-4-519-531 SN - 1997-1397 SN - 2313-6022 VL - 11 IS - 4 SP - 519 EP - 531 PB - Siberian Federal University CY - Krasnoyarsk ER - TY - JOUR A1 - Bagderina, Yulia Yu. A1 - Tarkhanov, Nikolai Nikolaevich T1 - Solution of the equivalence problem for the third Painleve equation JF - Journal of mathematical physics N2 - We find necessary conditions for a second order ordinary differential equation to be equivalent to the Painleve III equation under a general point transformation. Their sufficiency is established by reduction to known results for the equations of the form y ' = f (x, y). We consider separately the generic case and the case of reducibility to an autonomous equation. The results are illustrated by the primary resonance equation. Y1 - 2015 U6 - https://doi.org/10.1063/1.4905383 SN - 0022-2488 SN - 1089-7658 VL - 56 IS - 1 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Elin, Mark A1 - Shoikhet, David A1 - Tarkhanov, Nikolai Nikolaevich T1 - Separation of boundary singularities for holomorphic generators JF - Annali di matematica pura ed applicata N2 - We prove a theorem on separation of boundary null points for generators of continuous semigroups of holomorphic self-mappings of the unit disk in the complex plane. Our construction demonstrates rather strikingly the particular role of the binary operation au broken vertical bar given by 1/ f au broken vertical bar g = 1/f + 1/g on generators. KW - Semigroup KW - Holomorphic map KW - Unit disk KW - Angular derivatives Y1 - 2011 U6 - https://doi.org/10.1007/s10231-010-0165-y SN - 0373-3114 VL - 190 IS - 4 SP - 595 EP - 618 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Kiselev, Oleg A1 - Tarkhanov, Nikolai Nikolaevich T1 - Scattering of trajectories at a separatrix under autoresonance JF - Journal of mathematical physics N2 - The subject of this paper is solutions of an autoresonance equation. We look for a connection between the parameters of the solution bounded as t -> -infinity, and the parameters of two two-parameter families of solutions as t -> infinity. One family consists of the solutions which are not captured into resonance, and another of those increasing solutions which are captured into resonance. In this way we describe the transition through the separatrix for equations with slowly varying parameters and get an estimate for parameters before the resonance of those solutions which may be captured into autoresonance. (C) 2014 AIP Publishing LLC. Y1 - 2014 U6 - https://doi.org/10.1063/1.4875105 SN - 0022-2488 SN - 1089-7658 VL - 55 IS - 6 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Garifullinevich, Rustem Nail A1 - Suleimanov, Bulat Irekovich A1 - Tarkhanov, Nikolai Nikolaevich T1 - Phase shift in the Whitham zone for the Gurevich-Pitaevskii special solution of the Korteweg-de Vries equation N2 - We get the leading term of the Gurevich-Pitaevskii special solution of the KdV equation in the oscillation zone without using averaging methods. Y1 - 2010 UR - http://www.sciencedirect.com/science/journal/03759601 U6 - https://doi.org/10.1016/j.physleta.2010.01.057 SN - 0375-9601 ER -