arXiv subjects

K. Khanin

Publications and source records attributed to K. Khanin.

5 recordsLinked to original sources

arXiv2004

Multifractality of the Feigenbaum attractor and fractional derivatives

It is shown that fractional derivatives of the (integrated) invariant measure of the Feigenbaum map at the onset of chaos have power-law tails in their cumulative distributions, whose exponents can be related to the spectrum of singularities $f(α)$. This is a new way of characterizing multifractality in dynamical systems, so far applied only to multifractal random functions (Frisch and Matsumoto (J. Stat. Phys. 108:1181, 2002)). The relation between the thermodynamic approach (Vul, Sinai and Khanin (Russian Math. Surveys 39:1, 1984)) and that based on singularities of the invariant measures is also examined. The theory for fractional derivatives is developed from a heuristic point view and tested by very accurate simulations.

nlin.CD↗

arXiv2003

Forced Burgers Equation in an Unbounded Domain

The inviscid Burgers equation with random and spatially smooth forcing is considered in the limit when the size of the system tends to infinity. For the one-dimensional problem, it is shown both theoretically and numerically that many of the features of the space-periodic case carry over to infinite domains as intermediate time asymptotics. In particular, for large time $T$ we introduce the concept of $T$-global shocks replacing the notion of main shock which was considered earlier in the periodic case (1997, E et al., Phys. Rev. Lett. 78, 1904). In the case of spatially extended systems these objects are no anymore global. They can be defined only for a given time scale and their spatial density behaves as $ρ(T) \sim T^{-2/3}$ for large $T$. The probability density function $p(A)$ of the age $A$ of shocks behaves asymptotically as $A^{-5/3}$. We also suggest a simple statistical model for the dynamics and interaction of shocks and discuss an analogy with the problem of distribution of instability islands for a simple first-order stochastic differential equation.

nlin.CD↗

arXiv2002

Topological Shocks in Burgers Turbulence

The dynamics of the multi-dimensional randomly forced Burgers equation is studied in the limit of vanishing viscosity. It is shown both theoretically and numerically that the shocks have a universal global structure which is determined by the topology of the configuration space. This structure is shown to be particularly rigid for the case of periodic boundary conditions.

nlin.CD↗

arXiv2001

A universal law for tails of density pdf's in multi-dimensional Burgers turbulence

Extending work of E, Khanin, Mazel and Sinai (1997 PRL 78:1904-1907) on the one-dimensional Burgers equation, we show that density pdf's have universal power-law tails with exponent -7/2. This behavior stems from singularities, other than shocks, whose nature is quite different in one and several dimensions. We briefly discuss the possibility of detecting singularities of Navier-Stokes turbulence using pdf tails.

astro-ph↗

arXiv2000

Kicked Burgers Turbulence

Burgers turbulence subject to a force $f(x,t)=\sum_jf_j(x)δ(t-t_j)$, where the $t_j$'s are ``kicking times'' and the ``impulses'' $f_j(x)$ have arbitrary space dependence, combines features of the purely decaying and the continuously forced cases. With large-scale forcing this ``kicked'' Burgers turbulence presents many of the regimes proposed by E, Khanin, Mazel and Sinai (1997) for the case of random white-in-time forcing. It is also amenable to efficient numerical simulations in the inviscid limit, using a modification of the Fast Legendre Transform method developed for decaying Burgers turbulence by Noullez and Vergassola (1994). For the kicked case, concepts such as ``minimizers'' and ``main shock'', which play crucial roles in recent developments for forced Burgers turbulence, become elementary since everything can be constructed from simple two-dimensional area-preserving Euler--Lagrange maps. One key result is for the case of identical deterministic kicks which are periodic and analytic in space and are applied periodically in time: the probability densities of large negative velocity gradients and of (not-too-large) negative velocity increments follow the power law with -7/2 exponent proposed by E {\it et al}. (1997) in the inviscid limit, whose existence is still controversial in the case of white-in-time forcing. (More in the full-length abstract at the beginning of the paper.)

chao-dyn↗