24,665 research outputs found
On Krein-like theorems for noncanonical Hamiltonian systems with continuous spectra: application to Vlasov-Poisson
The notions of spectral stability and the spectrum for the Vlasov-Poisson
system linearized about homogeneous equilibria, f_0(v), are reviewed.
Structural stability is reviewed and applied to perturbations of the linearized
Vlasov operator through perturbations of f_0. We prove that for each f_0 there
is an arbitrarily small delta f_0' in W^{1,1}(R) such that f_0+delta f_0f_0$ is perturbed by an area preserving rearrangement, f_0 will
always be stable if the continuous spectrum is only of positive signature,
where the signature of the continuous spectrum is defined as in previous work.
If there is a signature change, then there is a rearrangement of f_0 that is
unstable and arbitrarily close to f_0 with f_0' in W^{1,1}. This result is
analogous to Krein's theorem for the continuous spectrum. We prove that if a
discrete mode embedded in the continuous spectrum is surrounded by the opposite
signature there is an infinitesimal perturbation in C^n norm that makes f_0
unstable. If f_0 is stable we prove that the signature of every discrete mode
is the opposite of the continuum surrounding it.Comment: Submitted to the journal Transport Theory and Statistical Physics. 36
pages, 12 figure
Temperature gradient and electric field driven electrostatic instabilities
The stability of electrostatic waves to thermodynamic and electric potential gradients was investigated. It is shown that thermodynamic gradients drive instabilities even when the internal electric field vanishes. Skewing of the distribution function is not included in the dielectric
Pattern formation in Hamiltonian systems with continuous spectra; a normal-form single-wave model
Pattern formation in biological, chemical and physical problems has received
considerable attention, with much attention paid to dissipative systems. For
example, the Ginzburg--Landau equation is a normal form that describes pattern
formation due to the appearance of a single mode of instability in a wide
variety of dissipative problems. In a similar vein, a certain "single-wave
model" arises in many physical contexts that share common pattern forming
behavior. These systems have Hamiltonian structure, and the single-wave model
is a kind of Hamiltonian mean-field theory describing the patterns that form in
phase space. The single-wave model was originally derived in the context of
nonlinear plasma theory, where it describes the behavior near threshold and
subsequent nonlinear evolution of unstable plasma waves. However, the
single-wave model also arises in fluid mechanics, specifically shear-flow and
vortex dynamics, galactic dynamics, the XY and Potts models of condensed matter
physics, and other Hamiltonian theories characterized by mean field
interaction. We demonstrate, by a suitable asymptotic analysis, how the
single-wave model emerges from a large class of nonlinear advection-transport
theories. An essential ingredient for the reduction is that the Hamiltonian
system has a continuous spectrum in the linear stability problem, arising not
from an infinite spatial domain but from singular resonances along curves in
phase space whereat wavespeeds match material speeds (wave-particle resonances
in the plasma problem, or critical levels in fluid problems). The dynamics of
the continuous spectrum is manifest as the phenomenon of Landau damping when
the system is ... Such dynamical phenomena have been rediscovered in different
contexts, which is unsurprising in view of the normal-form character of the
single-wave model
- …