LOcalized modes on an Ablowitz-Ladik nonlinear impurity

We study localized modes on a single Ablowitz-Ladik impurity embedded in the bulk or at the surface of a one-dimensional linear lattice. Exact expressions are obtained for the bound state profile and energy. Dynamical excitation of the localized mode reveals exponentially-high amplitude oscillations of the spatial profile at the impurity location. The presence of a surface increases the minimum nonlinearity to effect a dynamical selftrapping.Comment: 11 pages, 6 figures, accepted in PL

The nonlinear magnetoinductive dimer

We examine a nonlinear magnetoinductive dimer and compute its linear and nonlinear symmetric, antisymmetric and asymmetric modes in closed-form, in the rotating-wave approximation. A linear stability analysis of these modes reveals that the asymmetric mode is always stable, for any allowed value of the coupling parameter and for both, hard and soft nonlinearity. A numerical computation of the dimer dynamics reveals a magnetic energy selftrapping whose threshold increases for increasing dimer coupling.Comment: 4 double-column pages, 6 figures, submitte

Nonlinear Impurity in a Lattice: Dispersion Effects

We examine the bound state(s) associated with a single cubic nonlinear impurity, in a one-dimensional tight-binding lattice, where hopping to first--and--second nearest neighbors is allowed. The model is solved in closed form {\em v\`{\i}a} the use of the appropriate lattice Green function and a phase diagram is obtained showing the number of bound states as a function of nonlinearity strength and the ratio of second to first nearest--neighbor hopping parameters. Surprisingly, a finite amount of hopping to second nearest neighbors helps the formation of a bound state at smaller (even vanishingly small) nonlinearity values. As a consequence, the selftrapping transition can also be tuned to occur at relatively small nonlinearity strength, by this increase in the lattice dispersion.Comment: 24 pages, 10 figure

Flat bands and PT-symmetry in quasi-one-dimensional lattices

We examine the effect of adding PT-symmetric gain and loss terms to quasi 1D lattices (ribbons) that possess flat bands. We focus on three representative cases: (a) The Lieb ribbon, (b) The kagome ribbon, and (c) The stub Ribbon. In general we find that the effect on the flat band depends strongly on the geometrical details of the lattice being examined. One interesting and novel result that emerge from an analytical calculation of the band structure of the Lieb ribbon including gain and loss, is that its flat band survives the addition of PT-symmetry for any amount of gain and loss, while for the other two lattices, any presence of gain and loss destroys the flat bands. For all three ribbons, there are finite stability windows whose size decreases with the strength of the gain and loss parameter. For the Lieb and kagome cases, the size of this window converges to a finite value. The existence of finite stability windows, plus the constancy of the Lieb flat band are in marked contrast to the behavior of a pure one-dimensional lattice.Comment: 5 pages, 5 figure

Ideal Gas in a Finite Container

The thermodynamics of an ideal gas enclosed in a box of volume a1 x a2 x a3 at temperature T is considered. The canonical partition function of the system is expressed in terms of complete elliptic integrals of the first kind, whose argument obeys a transcendental equation. For high and low temperatures we derive explicitly the main finite-volume corrections to the standard thermodynamic quantities.Comment: 13 pages total (Latex source), including one table and one ps figur

Exponential versus linear amplitude decay in damped oscillators

We comment of the widespread belief among some undergraduate students that the amplitude of any harmonic oscillator in the presence of any type of friction, decays exponentially in time. To dispel that notion, we compare the amplitude decay for a harmonic oscillator in the presence of (i) viscous friction and (ii) dry friction. It is shown that, in the first case, the amplitude decays exponentially with time while in the second case, it decays linearly with time.Comment: 3 pages, 1 figure, accepted in Phys. Teac