278 research outputs found
Stability of discrete dark solitons in nonlinear Schrodinger lattices
We obtain new results on the stability of discrete dark solitons bifurcating
from the anti-continuum limit of the discrete nonlinear Schrodinger equation,
following the analysis of our previous paper [Physica D 212, 1-19 (2005)]. We
derive a criterion for stability or instability of dark solitons from the
limiting configuration of the discrete dark soliton and confirm this criterion
numerically. We also develop detailed calculations of the relevant eigenvalues
for a number of prototypical configurations and obtain very good agreement of
asymptotic predictions with the numerical data.Comment: 11 pages, 5 figure
Finite-time singularities in the dynamical evolution of contact lines
We study finite-time singularities in the linear advection-diffusion equation
with a variable speed on a semi-infinite line. The variable speed is determined
by an additional condition at the boundary, which models the dynamics of a
contact line of a hydrodynamic flow at a 180 contact angle. Using apriori
energy estimates, we derive conditions on variable speed that guarantee that a
sufficiently smooth solution of the linear advection--diffusion equation blows
up in a finite time. Using the class of self-similar solutions to the linear
advection-diffusion equation, we find the blow-up rate of singularity
formation. This blow-up rate does not agree with previous numerical simulations
of the model problem.Comment: 9 pages, 2 figure
Periodic oscillations of dark solitons in parabolic potentials
We reformulate the Gross-Pitaevskii equation with an external parabolic
potential as a discrete dynamical system, by using the basis of Hermite
functions. We consider small amplitude stationary solutions with a single node,
called dark solitons, and examine their existence and linear stability.
Furthermore, we prove the persistence of a periodic motion in a neighborhood of
such solutions. Our results are corroborated by numerical computations
elucidating the existence, linear stability and dynamics of the relevant
solutions.Comment: 20 pages, 3 figure
PT-symmetric lattices with spatially extended gain/loss are generically unstable
We illustrate, through a series of prototypical examples, that linear
parity-time (PT) symmetric lattices with extended gain/loss profiles are
generically unstable, for any non-zero value of the gain/loss coefficient. Our
examples include a parabolic real potential with a linear imaginary part and
the cases of no real and constant or linear imaginary potentials. On the other
hand, this instability can be avoided and the spectrum can be real for
localized or compact PT-symmetric potentials. The linear lattices are analyzed
through discrete Fourier transform techniques complemented by numerical
computations.Comment: 6 pages, 4 figure
Discrete solitons in PT-symmetric lattices
We prove existence of discrete solitons in infinite parity-time (PT-)
symmetric lattices by means of analytical continuation from the anticontinuum
limit. The energy balance between dissipation and gain implies that in the
anticontinuum limit the solitons are constructed from elementary PT-symmetric
blocks such as dimers, quadrimers, or more general oligomers. We consider in
detail a chain of coupled dimers, analyze bifurcations of discrete solitons
from the anticontinuum limit and show that the solitons are stable in a
sufficiently large region of the lattice parameters. The generalization of the
approach is illustrated on two examples of networks of quadrimers, for which
stable discrete solitons are also found.Comment: 6 pages, 6 figures; accepted to EPL, www.epletters.ne
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