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

Approximation of small-amplitude weakly coupled oscillators with discrete nonlinear Schrodinger equations

Small-amplitude weakly coupled oscillators of the Klein-Gordon lattices are approximated by equations of the discrete nonlinear Schrodinger type. We show how to justify this approximation by two methods, which have been very popular in the recent literature. The first method relies on a priori energy estimates and multi-scale decompositions. The second method is based on a resonant normal form theorem. We show that although the two methods are different in the implementation, they produce equivalent results as the end product. We also discuss applications of the discrete nonlinear Schrodinger equation in the context of existence and stability of breathers of the Klein--Gordon lattice

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

Vortex families near a spectral edge in the Gross-Pitaevskii equation with a two-dimensional periodic potential

We examine numerically vortex families near band edges of the Bloch wave spectrum in the Gross--Pitaevskii equation with a two-dimensional periodic potential and in the discrete nonlinear Schroedinger equation. We show that besides vortex families that terminate at a small distance from the band edges via fold bifurcations there exist vortex families that are continued all way to the band edges.Comment: 12 pages, 8 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