Nonlinear Impurity Research Articles

Trapping of discrete solitons by defects in nonlinear waveguide arrays

We study the trapping process of moving discrete solitons by linear and nonlinear impurities embedded in a one-dimensional nonlinear cubic array. We show that there exist specific values for the strength of the impurity and for the angle where a strong trapping is obtained. We introduce a criterion for studying the scattering dynamics of localized waves in nonlinear extended systems where the trapping of energy takes place.

Interaction of a discrete soliton with a surface mode

We examine the formation of bound state(s) near the surface of a semi-infinite tight-binding chain, when both, a linear surface mode and a nonlinear impurity located at distance $d$ from the surface, are present. By using the formalism of lattice Green functions, we obtain an exact equation for the allowed bound states, and find that in general, up to five bound states are possible, although not all of them are stable. The presence of a surface mode can alter considerably the critical nonlinearity needed to form a bound state on the nonlinear impurity site, when the impurity is close to the surface. When the surface state is ``on,'' interference effects between its tail and the discrete soliton tail can lead to a destabilization of the latter.

Impurity modes and wave scattering in discrete chains with nonlinear defect states

We study the properties of localized impurity modes and the transmission of plane waves across a general nonlinear impurity embedded in an infinite one-dimensional linear chain. Using the formalism of the lattice Green function, we obtain nonlinear equations for the bound state energy and the transmission coefficient, for a wide class of nonlinear impurities. We specialize to a saturable nonlinear impurity, obtaining closed form expressions for the bound state energy and transmission coefficient.

Nonlinear surface impurity in a semi-infinite lattice

We examine the formation of bound states on a generalized nonlinear impurity located at or near the beginning (surface) of a linear, tight-binding semi-infinite lattice. Using the formalism of lattice Green functions, we obtain in closed form the number of bound states as well as their energies and probability profiles, for different nonlinearity parameter values and nonlinearity exponents, at different distances from the surface. It is shown that close to the surface, the amount of nonlinearity needed to create a bound state or to effect dynamical selftrapping, increases (decreases) depending on whether the exponent is smaller (larger) than, approximately, two.

Charging effect on transport through a nonlinear impurity

We use a generalized discrete nonlinear Schrödinger (DNLS) equation to describe tunnelling through a single magnetic impurity connected to two perfect leads, in the presence of a magnetic field. The charge build up at the impurity site is taken into account through the use of a self consistent Hartree approach. The transmission and conductance through the localized impurity is being considered in the presence of both local nonlinear interaction and a magnetic field which lifts the spin degeneracy at the impurity site.

Instabilities and bifurcations of nonlinear impurity modes.

We study the structure and stability of nonlinear impurity modes in the discrete nonlinear Schrödinger equation with a single on-site nonlinear impurity emphasizing the effects of interplay between discreteness, nonlinearity, and disorder. We show how the interaction of a nonlinear localized mode (a discrete soliton or discrete breather) with a repulsive impurity generates a family of stationary states near the impurity site, as well as examine both theoretical and numerical criteria for the transition between different localized states via a cascade of bifurcations.

NEW DNLS EQUATIONS FOR ANHARMONIC VIBRATIONAL IMPURITIES

We examine some new DNLS-like equations that arise when considering strongly-coupled electron-vibration systems, where the local oscillator potential is anharmonic. In particular, we focus on a single, rather general nonlinear vibrational impurity and determine its bound state(s) and its dynamical selftrapping properties.

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.

Conductance through a single nonlinear impurity

We use the discrete nonlinear Schrödinger (DNLS) equation to describe tunneling through a single magnetic impurity connected to two perfect leads, in the presence of a magnetic field. Due to the presence of a strong on-site nonlinear interaction between two opposite spins at the impurity site, the zero voltage conductance exhibits strong correlations between parallel and anti-parallel spin conduction channels.

Self-trapping transition for nonlinear impurities embedded in a Cayley tree

The self-trapping transition due to a single and a dimer nonlinear impurity embedded in a Cayley tree is studied. In particular, the effect of a perfectly nonlinear Cayley tree is considered. A sharp self-trapping transition is observed in each case. It is also observed that the transition is much sharper compared to the case of one-dimensional lattices. For each system, the critical values of $\chi$ for the self-trapping transitions are found to obey a power-law behavior as a function of the connectivity $K$ of the Cayley tree.

Nonlinearity and disorder: classification and stability of nonlinear impurity modes.

We study the effects produced by competition of two physical mechanisms of energy localization in inhomogeneous nonlinear systems. As an example, we analyze spatially localized modes supported by a nonlinear impurity in the generalized nonlinear Schrödinger equation and describe three types of nonlinear impurity modes, one- and two-hump symmetric localized modes and asymmetric localized modes, for both focusing and defocusing nonlinearity and two different (attractive or repulsive) types of impurity. We obtain an analytical stability criterion for the nonlinear localized modes and consider the case of a power-law nonlinearity in detail. We discuss several scenarios of the instability-induced dynamics of the nonlinear impurity modes, including the mode decay or switching to a new stable state, and collapse at the impurity site.

Universal features of self-trapping in nonlinear tight-binding lattices

We show that nonlinear tight-binding lattices of different geometries and dimensionalities, display an universal selftrapping behavior. First, we consider the single nonlinear impurity problem in various tight-binding lattices, and use the Green's function formalism for an exact calculation of the minimum nonlinearity strength to form a stationary bound state. For all lattices, we find that this critical nonlinearity parameter (scaled by the energy of the bound state), in terms of the nonlinearity exponent, falls inside a narrow band, which converges to e^(1/2) at large exponent values. Then, we use the Discrete Nonlinear Schroedinger (DNLS) equation to examine the selftrapping dynamics of a single excitation, initially localized on the single nonlinear site, and compute the critical nonlinearity parameter for abrupt dynamical selftrapping. For a given nonlinearity exponent, this critical nonlinearity, properly scaled, is found to be nearly the same for all lattices. Same results are obtained when generalizing to completely nonlinear lattices, suggesting an underlying selftrapping universality behavior for all nonlinear (even disordered) tight-binding lattices described by DNLS.

Path-integral approach to strongly nonlinear composites

We study strongly nonlinear disordered media using a functional method. We solve exactly the problem of a nonlinear impurity in a linear host and we obtain a Bruggeman-like formula for the effective nonlinear susceptibility. This formula reduces to the usual Bruggeman effective medium approximation in the linear case and has the following features: (i) It reproduces the weak contrast expansion to the second order and (ii) the effective medium exponent near the percolation threshold are $s=1$, $t=1+\kappa$, where $\kappa$ is the nonlinearity exponent. Finally, we give analytical expressions for previously numerically calculated quantities.

Stationary localized states due to nonlinear impurities described by the modified discrete nonlinear Schrödinger equation

The modified discrete nonlinear Schrödinger equation is used to study the formation of stationary localized states in a one-dimensional lattice with a single impurity and an asymmetric dimer impurity. A periodically modulated nonlinear chain and a perfectly nonlinear chain are also considered. Phase diagrams of localized states for all the systems and stability of the states are discussed.

Pulse-driven switching in one-dimensional nonlinear photonic band gap materials: a numerical study

We numerically examine the time-dependent properties of nonlinear bistable multilayer structures for constant wave illumination. We find that our system exhibits both steady-state and self-pulsing solutions. In the steady-state regime, we examine the dynamics of driving the system between different transmission states by injecting pulses, and we find optimal pulse parameters. We repeat this work for the case of a linear periodic system with a nonlinear impurity layer.

The transmittance pattern in a mesoscopic nonuniform necklace like system

The energy spectrum, the density of states, and the electronic transmittance pattern of a necklace like chain under the influence of magnetic field (uniform and nonuniform) and nonlinear impurity scattering (ordered and disordered) are discussed. A variety of quantum interference effects exhibited in the transmittance pattern are discussed, and physical explanations of these effects and their applications in designing quantum devices are analyzed.

Stationary localized states in one-dimensional systems due to modified nonlinear impurities

Discrete nonlinear Schr\"odinger equation (DNLS) of the form ${i(dC}_{n}{/dt)=C}_{n+1}{+C}_{n\ensuremath{-}1}\ensuremath{-}{\ensuremath{\chi}}_{n}(|{C}_{n+1}{|}^{2}+|{C}_{n\ensuremath{-}1}{|}^{2}\ensuremath{-}2|{C}_{n}{|}^{2}{)C}_{n}$ is used to study the formation of stationary localized states in one dimensional system due to a single as well as a dimeric nonlinear impurity. The stability of the states and its connection with the nonlinear strength is presented. Results are compared with those obtained from other DNLS's. It is found that the DNLS used in this paper has more impact on the formation of stationary localized states.

Intrinsic localized modes in nonlinear photonic crystal waveguides: Dispersive modes

A discussion is given of the theory of intrinsic localized modes (ILM) for nonlinear waveguide impurities in photonic crystals. Particular attention is made to the determination of the dispersion relation of propagating ILM modes. Both even- and odd-parity ILM modes are considered.

Nonlinear impurity in a square lattice

We use the Green's-function formalism for an exact, numerical calculation of the stationary states of an electron propagating in a square lattice in the presence of a single, Holstein-type, impurity of arbitrary nonlinearity exponent. We find that two bound states exist above a certain exponent-dependent critical nonlinearity strength. The localization length of the lower (higher) energy bound state increases (decreases) with nonlinearity strength. The dynamics of an electron, initially placed on the impurity site, reveals a sharp, self-trapping transition for any nonzero nonlinearity exponent: below a certain nonlinearity threshold, the electron escapes from the impurity site ballistically; above the threshold, there is partial trapping at the impurity site while the untrapped fraction escapes to infinity, also ballistically. The self-trapping features are sharper in time and space than for its one-dimensional analogue.

Intrinsic localized modes in nonlinear photonic crystal waveguides

A discussion is given of the theory of intrinsic localized modes (ILM) for nonlinear waveguide impurities in photonic crystals. Conditions are determined for the existence of both even- and odd-parity ILM modes and kink ILM modes. A study is also made of ILM modes in a two-dimensional array of Kerr impurities.