1,198 research outputs found
Formation of singularities on the surface of a liquid metal in a strong electric field
The nonlinear dynamics of the free surface of an ideal conducting liquid in a
strong external electric field is studied. It is establish that the equations
of motion for such a liquid can be solved in the approximation in which the
surface deviates from a plane by small angles. This makes it possible to show
that on an initially smooth surface for almost any initial conditions points
with an infinite curvature corresponding to branch points of the root type can
form in a finite time.Comment: 14 page
First direct measurement of optical phonons in 2D plasma crystals
Spectra of phonons with out-of-plane polarization were studied experimentally
in a 2D plasma crystal. The dispersion relation was directly measured for the
first time using a novel method of particle imaging. The out-of-plane mode was
proven to have negative optical dispersion, comparison with theory showed good
agreement. The effect of the plasma wakes on the dispersion relation is briefly
discussed.Comment: submitted to Physical Review Letter
Wave mode coupling due to plasma wakes in two-dimensional plasma crystals: In-depth view
Experiments with two-dimensional (2D) plasma crystals are usually carried out
in rf plasma sheaths, where the interparticle interactions are modified due to
the presence of plasma wakes. The wake-mediated interactions result in the
coupling between wave modes in 2D crystals, which can trigger the mode-coupling
instability and cause melting. The theory predicts a number of distinct
fingerprints to be observed upon the instability onset, such as the emergence
of a new hybrid mode, a critical angular dependence, a mixed polarization, and
distinct thresholds. In this paper we summarize these key features and provide
their detailed discussion, analyze the critical dependence on experimental
parameters, and highlight the outstanding issues
On separable Fokker-Planck equations with a constant diagonal diffusion matrix
We classify (1+3)-dimensional Fokker-Planck equations with a constant
diagonal diffusion matrix that are solvable by the method of separation of
variables. As a result, we get possible forms of the drift coefficients
providing separability of the
corresponding Fokker-Planck equations and carry out variable separation in the
latter. It is established, in particular, that the necessary condition for the
Fokker-Planck equation to be separable is that the drift coefficients must be linear. We also find the necessary condition for
R-separability of the Fokker-Planck equation. Furthermore, exact solutions of
the Fokker-Planck equation with separated variables are constructedComment: 20 pages, LaTe
Direct observation of mode-coupling instability in two-dimensional plasma crystals
Dedicated experiments on melting of 2D plasma crystals were carried out. The
melting was always accompanied by spontaneous growth of the particle kinetic
energy, suggesting a universal plasma-driven mechanism underlying the process.
By measuring three principal dust-lattice (DL) wave modes simultaneously, it is
unambiguously demonstrated that the melting occurs due to the resonance
coupling between two of the DL modes. The variation of the wave modes with the
experimental conditions, including the emergence of the resonant (hybrid)
branch, reveals exceptionally good agreement with the theory of mode-coupling
instability.Comment: 4 pages, submitted to Physical Review Letter
Nonlinear regime of the mode-coupling instability in 2D plasma crystals
The transition between linear and nonlinear regimes of the mode-coupling
instability (MCI) operating in a monolayer plasma crystal is studied. The mode
coupling is triggered at the centre of the crystal and a melting front is
formed, which travels through the crystal. At the nonlinear stage, the mode
coupling results in synchronisation of the particle motion and the kinetic
temperature of the particles grows exponentially. After melting of the
crystalline structure, the mean kinetic energy of the particles continued to
grow further, preventing recrystallisation of the melted phase. The effect
could not be reproduced in simulations employing a simple point-like wake
model. This shows that at the nonlinear stage of the MCI a heating mechanism is
working which was not considered so far.Comment: 6 pages, 4 figure
Group classification of heat conductivity equations with a nonlinear source
We suggest a systematic procedure for classifying partial differential
equations invariant with respect to low dimensional Lie algebras. This
procedure is a proper synthesis of the infinitesimal Lie's method, technique of
equivalence transformations and theory of classification of abstract low
dimensional Lie algebras. As an application, we consider the problem of
classifying heat conductivity equations in one variable with nonlinear
convection and source terms. We have derived a complete classification of
nonlinear equations of this type admitting nontrivial symmetry. It is shown
that there are three, seven, twenty eight and twelve inequivalent classes of
partial differential equations of the considered type that are invariant under
the one-, two-, three- and four-dimensional Lie algebras, correspondingly.
Furthermore, we prove that any partial differential equation belonging to the
class under study and admitting symmetry group of the dimension higher than
four is locally equivalent to a linear equation. This classification is
compared to existing group classifications of nonlinear heat conductivity
equations and one of the conclusions is that all of them can be obtained within
the framework of our approach. Furthermore, a number of new invariant equations
are constructed which have rich symmetry properties and, therefore, may be used
for mathematical modeling of, say, nonlinear heat transfer processes.Comment: LaTeX, 51 page
Group classification of (1+1)-Dimensional Schr\"odinger Equations with Potentials and Power Nonlinearities
We perform the complete group classification in the class of nonlinear
Schr\"odinger equations of the form
where is an arbitrary
complex-valued potential depending on and is a real non-zero
constant. We construct all the possible inequivalent potentials for which these
equations have non-trivial Lie symmetries using a combination of algebraic and
compatibility methods. The proposed approach can be applied to solving group
classification problems for a number of important classes of differential
equations arising in mathematical physics.Comment: 10 page
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