arXiv subjects

Ilia G. Shukhman

Publications and source records attributed to Ilia G. Shukhman.

8 recordsLinked to original sources

arXiv2026

A generalized linear matrix method for normal modes in collisionless stellar disks

We generalize the linear matrix method for computing normal modes in collisionless stellar disks to distribution functions with sharp edges at zero angular momentum ($L=0$). The generalization adds boundary-integral terms to the matrix equation without increasing its size. We validate the method by computing $m=2$ modes for two Kuzmin--Toomre disk models (Miyamoto $n_{\rm M}=3$ and Kalnajs $m_{\rm K}=6$ families) and comparing the eigenvalues with those obtained from an independent nonlinear matrix method based on logarithmic-spiral expansions. A systematic convergence study over grid resolution and harmonic truncation yields eigenvalues accurate to ${\sim}\,0.003$ in both pattern speed and growth rate. Unlike the nonlinear method, the linear method naturally incorporates gravitational softening, enabling the computation of eigenmodes for softened disk models. The implementation in Julia with GPU acceleration is openly available.

astro-ph.GA↗

arXiv2026

Landau damping of disturbances in nearly inviscid inflectional shear flows

We investigate the structure of damped two-dimensional perturbations in unstable plane-parallel shear flows with an inflection point. In inviscid flows within the stable wavenumber region $k$, no regular eigenmodes exist -- the frequency spectrum $ω$ consists of a continuous set of singular van Kampen modes with real frequencies. Nevertheless, initial perturbations of the total vorticity integrated across the flow decay exponentially, resembling the behavior of an eigenmode with complex eigenfrequency ${\rm Im}\,ω<0$ (Landau damping). However, the vorticity itself does not decay but becomes increasingly corrugated across the flow. We demonstrate that accounting for arbitrarily small viscosity transforms this exponentially decaying perturbation into a true eigenmode in which the vorticity preserves its spatial form. We numerically trace the transformation of the vorticity structure of this mode and its disappearance as viscosity approaches zero. We discuss similarities and differences between the behavior of damped perturbations in the transition from inviscid to nearly inviscid flows in hydrodynamics and their behavior in plasma and homogeneous stellar systems during the analogous transition from collisionless to very weakly collisional systems.

physics.flu-dyn↗

arXiv2025

Eigenvalues and Eigenfunctions of Landau Damping Oscillations in Very Weakly Collisional Plasma

Landau-damped oscillations in collisionless plasmas, described by van Kampen and Case, are quasi-modes, representing a continuous superposition of singular eigenfunctions, not true eigenmodes. Recent work by Ng et al. shows that even rare collisions replace these singular modes with discrete regular modes having complex eigenvalues for the phase velocity (or frequency), approaching Landau eigenvalues in the collisionless limit. We analytically derive approximate expressions for the eigenvalue correction due to rare collisions and for the shape of the eigenfunction describing DF perturbations in velocity space, demonstrating its increasing oscillations in the resonance region as the collision frequency tends to zero. We also obtain approximate expressions for the resonance region's width and peak value, and the oscillation period within it. We validate these analytical results with high-precision numerical calculations using a standard linear matrix eigenvalue problem approach.

physics.plasm-ph↗

arXiv2025

Dynamics of stellar systems with collisions: eigenvalues and eigenfunctions in nearly collisionless limit

We examine the decay of perturbations in an infinite homogeneous self-gravitating model with a Maxwellian distribution function (DF) when weak collisions are present. In collisionless systems within the stable parameter range, the eigenvalue spectrum consists of a continuous set of real frequencies associated with van Kampen modes, which are singular eigenfunctions of the stellar DF. An initial perturbation in the stellar density and gravitational potential decays exponentially through a superposition of these modes, a phenomenon known as Landau damping. However, the perturbation in the stellar DF does not decay self-similarly; it becomes increasingly oscillatory in velocity space over time, indicating the absence of eigenfunctions corresponding to the Landau damping eigenfrequencies. Consequently, we refer to perturbations undergoing Landau damping as quasi-modes rather than true eigenmodes. Even rare collisions suppress the formation of steep DF gradients in velocity space. C. S. Ng & A. Bhattacharjee demonstrated that introducing collisions eliminates van Kampen modes and transforms Landau quasi-modes into true eigenmodes forming a complete set. As the collision frequency approaches zero, their eigenfrequencies converge to those of the collisionless Landau quasi-modes. In this study, we investigate the behavior of the eigenfunction of the least-damped aperiodic mode as the collision frequency approaches zero. We derive analytic expressions for the eigenfunction in the resonance region and for the damping rate as a function of collision frequency. Additionally, we employ the standard matrix eigenvalue problem approach to numerically verify our analytical results.

astro-ph.GA↗

arXiv2023

Scale-invariant Mode in Collisionless Spherical Stellar Systems

An analytical solution of the perturbed equations is obtained, which exists in all ergodic models of collisionless spherical stellar systems with a single length parameter. This solution corresponds to variations of this parameter, i.e., stretching or shrinking the sphere while preserving the total mass. The system remains in an equilibrium state. The simplicity of the solution allows for explicit expressions for the distribution function, potential, and density at all orders of perturbation theory. This, in turn, helps to clarify the concept of perturbation energy, which, being a second-order quantity in amplitude, cannot be calculated in linear theory. It is shown that the correct expression for perturbation energy, constructed taking into account 2nd order perturbations, and the well-known expression for perturbation energy constructed as bilinear form obtained within linear theory from 1st order perturbations do not coincide. However, both of these energies are integrals of motion and differ only by a constant. The obtained solution can be used to control the correctness of codes and the accuracy of calculations in the numerical study of collisionless stellar models.

astro-ph.GA↗

arXiv2021

Damped perturbations in stellar systems: Genuine modes and Landau-damped waves

This research was stimulated by the recent studies of damping solutions in dynamically stable spherical stellar systems. Using the simplest model of the homogeneous stellar medium, we discuss nontrivial features of stellar systems. Taking them into account will make it possible to correctly interpret the results obtained earlier and will help to set up decisive numerical experiments in the future. In particular, we compare the initial value problem versus the eigenvalue problem. It turns out that in the unstable regime, the Landau-damped waves can be represented as a superposition of van Kampen modes {\it plus} a discrete damped mode, usually ignored in the stability study. This mode is a solution complex conjugate to the unstable Jeans mode. In contrast, the Landau-damped waves are not genuine modes: in modes, eigenfunctions depend on time as $\exp (-{\rm i} ωt)$, while the waves do not have eigenfunctions on the real $v$-axis at all. However, `eigenfunctions' on the complex $v$-contours do exist. Deviations from the Landau damping are common and can be due to singularities or cut-off of the initial perturbation above some fixed value in the velocity space.

astro-ph.GA↗

arXiv2018

Effect of Inner Lindblad Resonance on Spiral Density Waves Propagation in Disc Galaxies: Reflection over Absorption

Interaction of spiral density waves with stars in the vicinity of the inner Lindblad resonance in galactic discs is investigated using the linear perturbation theory and the leading orders in the epicyclic and WKB approximations. In analogy with shear flows in hydrodynamics, we conjecture that a weak nonlinearity in a narrow resonance region modifies the standard (Landau-Lin) bypass rule of the singularity to the integration in the principal value sense. This indeed leads to the reflection of the spiral wave instead of absorption, but the detailed picture looks awkward: the intervals of the wave weakening alternate with the intervals of the wave growth, so that the net absorption is absent. Incidentally, we rectify the result concerning leading spiral waves obtained earlier for the standard bypass rule.

astro-ph.GA↗

arXiv2007

Effect of angular momentum distribution on gravitational loss-cone instability in stellar clusters around massive BH

Small perturbations in spherical and thin disk stellar clusters surrounding massive a black hole are studied. Due to the black hole, stars with sufficiently low angular momentum escape from the system through the loss cone. We show that stability properties of spherical clusters crucially depend on whether the distribution of stars is monotonic or non-monotonic in angular momentum. It turns out that only non-monotonic distributions can be unstable. At the same time the instability in disk clusters is possible for both types of distributions.

astro-ph↗