Volume 24, Number 6
Volume 24, Number 6, 2019
Chierchia L., Koudjinan C.
V. I. Arnold's “Pointwise” KAM Theorem
Abstract
We review V.I. Arnold's 1963 celebrated paper [1] Proof of A.N. Kolmogorov's Theorem on the Conservation of Conditionally Periodic Motions with a Small Variation in the Hamiltonian, and prove that, optimising Arnold's scheme, one can get ''sharp'' asymptotic quantitative conditions (as $\varepsilon \to 0$, $\varepsilon$ being the strength of the perturbation). All constants involved are explicitly computed.

Kudryashov N. A., Safonova D. V., Biswas A.
Painlevé Analysis and a Solution to the Traveling Wave Reduction of the Radhakrishnan – Kundu – Lakshmanan Equation
Abstract
This paper considers the Radhakrishnan – Kundu – Laksmanan (RKL) equation to
analyze dispersive nonlinear waves in polarizationpreserving fibers. The Cauchy problem for
this equation cannot be solved by the inverse scattering transform (IST) and we look for exact
solutions of this equation using the traveling wave reduction. The Painlevé analysis for the
traveling wave reduction of the RKL equation is discussed. A first integral of traveling wave
reduction for the RKL equation is recovered. Using this first integral, we secure a general
solution along with additional conditions on the parameters of the mathematical model. The
final solution is expressed in terms of the Weierstrass elliptic function. Periodic and solitary
wave solutions of the RKL equation in the form of the traveling wave reduction are presented
and illustrated.

Krajňák V., Ezra G. S., Wiggins S.
Roaming at Constant Kinetic Energy: Chesnavich's Model and the Hamiltonian Isokinetic Thermostat
Abstract
We consider the roaming mechanism for chemical reactions under the nonholonomic
constraint of constant kinetic energy. Our study is carried out in the context of the Hamiltonian
isokinetic thermostat applied to Chesnavich’s model for an ionmolecule reaction. Through an
analysis of phase space structures we show that imposing the nonholonomic constraint does
not prevent the system from exhibiting roaming dynamics, and that the origin of the roaming
mechanism turns out to be analogous to that found in the previously studied Hamiltonian case.

Karabanov A. A., Morozov A. D.
On Resonances in Hamiltonian Systems with Three Degrees of Freedom
Abstract
We address the dynamics of nearintegrable Hamiltonian systems with 3 degrees
of freedom in extended vicinities of unperturbed resonant invariant Liouville tori. The main
attention is paid to the case where the unperturbed torus satisfies two independent resonance
conditions. In this case the average dynamics is 4dimensional, reduced to a generalised
motion under a conservative force on the 2torus and is generically nonintegrable. Methods of
differential topology are applied to full description of equilibrium states and phase foliations of
the average system. The results are illustrated by a simple model combining the nondegeneracy
and nonintegrability of the isoenergetically reduced system.

Gebhard B., Ortega R.
Stability of Periodic Solutions of the $N$vortex Problem in General Domains
Abstract
We investigate stability properties of a type of periodic solutions of the $N$vortex problem on general domains $\Omega\subset \mathbb{R}^2$. The solutions in question bifurcate from rigidly rotating configurations of the wholeplane vortex system and a critical point $a_0\in\Omega$ of the Robin function associated to the Dirichlet Laplacian of $\Omega$. Under a linear stability condition on the initial rotating configuration, which can be verified for
examples
consisting of up to 4 vortices, we show that the linear stability of the induced solutions is solely determined by the type of the critical point $a_0$. If $a_0$ is a saddle, they are unstable. If $a_0$ is a nondegenerate maximum or minimum, they are stable in a certain linear sense. Since nondegenerate minima exist generically, our results apply to most domains $\Omega$. The influence of the general domain $\Omega$ can be seen as a perturbation breaking the symmetries of the $N$vortex system on $\mathbb{R}^2$. Symplectic reduction is not applicable and our analysis on linearized stability relies on the notion of approximate eigenvectors. Beyond linear stability, Herman's last geometric theorem allows us to prove the existence of isoenergetically orbitally stable solutions in the case of $N=2$ vortices.

Allilueva A. I., Shafarevich A. I.
Conic Lagrangian Varieties and Localized Asymptotic Solutions of Linearized Equations of Relativistic Gas Dynamics
Abstract
We study asymptotic solution of the Cauchy problem for the linearized system of
relativistic gas dynamics. We assume that initial condiditiopns are strongly localized near a
spacelike surface in the Minkowsky space. We prove that the solution can be decomposed into
three modes, corresponding to different routsb of the equations of characteristics. One of these
roots is twice degenerate and the there are no focal points in the corresponding miode. The other
two roots are simple; in order to describe the corresponding modes, we use the modificication
of the Maslov’s canonical operator which was obtained recently.

Bolotin S. V.
Jumps of Energy Near a Homoclinic Set of a Slowly Time Dependent Hamiltonian System
Abstract
We consider a Hamiltonian system depending on a parameter which slowly changes with rate $\varepsilon \ll 1$. If trajectories of the frozen autonomous system are periodic, then the system has adiabatic invariant which changes much slower than energy. For a system with 1 degree of freedom and a figure 8 separatrix, Anatoly Neishtadt [18] showed that for trajectories
crossing the separatrix, the adiabatic invariant, and hence the energy, have quasirandom jumps of order $\varepsilon$.
We prove a partial analog of Neishtadt's result for a system with $n$ degrees of freedom
such that the frozen system has a hyperbolic equilibrium possessing several homoclinic orbits.
We construct trajectories staying near the homoclinic set with energy having jumps of order $\varepsilon$ at time intervals of order $\ln\varepsilon$, so the energy may grow with rate $\varepsilon/\ln\varepsilon$. Away from the homoclinic set faster energy growth is possible: if the frozen system has chaotic behavior, Gelfreich and Turaev [16] constructed trajectories with energy growth rate of order $\varepsilon$.

Dobrokhotov S. Y., Minenkov D. S., Neishtadt A. I., Shlosman S. B.
Classical and Quantum Dynamics of a Particle in a Narrow Angle
Abstract
We consider the 2D Schrödinger equation with variable potential in the narrow
domain diffeomorphic to the wedge with the Dirichlet boundary condition. The corresponding
classical problem is the billiard in this domain. In general, the corresponding dynamical system is
not integrable. The small angle is a small parameter which allows one to make the averaging and
reduce the classical dynamical system to an integrable one modulo exponential small correction.
We use the quantum adiabatic approximation (operator separation of variables) to construct the
asymptotic eigenfunctions (quasimodes) of the Schr¨odinger operator. We discuss the relation
between classical averaging and constructed quasimodes. The behavior of quasimodes in the
neighborhood of the cusp is studied. We also discuss the relation between Bessel and Airy
functions that follows from different representations of asymptotics near the cusp.

Bolotov D. I., Bolotov M. I., Smirnov L. A., Osipov G. V., Pikovsky A.
Twisted States in a System of Nonlinearly Coupled Phase Oscillators
Abstract
We study the dynamics of the ring of identical phase oscillators with nonlinear
nonlocal coupling. Using the Ott – Antonsen approach, the problem is formulated as a system
of partial derivative equations for the local complex order parameter. In this framework, we
investigate the existence and stability of twisted states. Both fully coherent and partially
coherent stable twisted states were found (the latter ones for the first time for identical
oscillators). We show that twisted states can be stable starting from a certain critical value
of the medium length, or on a length segment. The analytical results are confirmed with direct
numerical simulations in finite ensembles.

Kruglov V. P., Kuznetsov S. P.
Topaj – Pikovsky Involution in the Hamiltonian Lattice of Locally Coupled Oscillators
Abstract
We discuss the Hamiltonian model of an oscillator lattice with local coupling. The
Hamiltonian model describes localized spatial modes of nonlinear the Schrödinger equation
with periodic tilted potential. The Hamiltonian system manifests reversibility of the Topaj –
Pikovsky phase oscillator lattice. Furthermore, the Hamiltonian system has invariant manifolds
with asymptotic dynamics exactly equivalent to the Topaj – Pikovsky model. We examine the
stability of trajectories belonging to invariant manifolds by means of numerical evaluation of
Lyapunov exponents. We show that there is no contradiction between asymptotic dynamics
on invariant manifolds and conservation of phase volume of the Hamiltonian system. We
demonstrate the complexity of dynamics with results of numerical simulations.

Borisov A. V., Tsiganov A. V.
On the Chaplygin Sphere in a Magnetic Field
Abstract
We consider the possibility of using Dirac’s ideas of the deformation of Poisson
brackets in nonholonomic mechanics. As an example, we analyze the composition of external
forces that do no work and reaction forces of nonintegrable constraints in the model of
a nonholonomic Chaplygin sphere on a plane. We prove that, when a solenoidal field is
applied, the general mechanical energy, the invariant measure and the conformally Hamiltonian
representation of the equations of motion are preserved. In addition, we consider the case of
motion of the nonholonomic Chaplygin sphere in a constant magnetic field taking dielectric
and ferromagnetic (superconducting) properties of the sphere into account. As a byproduct
we also obtain two new integrable cases of the Hamiltonian rigid body dynamics in a constant
magnetic field taking the magnetization by rotation effect into account.
