Discrete Nonlinear Schrodinger Research Articles

Existence of localized solutions in the parametrically driven and damped DNLS equation in high-dimensional lattices

Instead of using the homoclinic orbit approach, which was commonly taken when studying the localized solutions of the discrete non-linear Schrödinger (DNLS) equation in one-dimensional lattices, we apply the continuation theorem to investigate the existence, stability, and spatial complexity of the localized solutions, including bright breathers, dark breathers, and anti-phase breathers, of the parametrically driven and damped DNLS equation in high-dimensional lattices. In particular, we prove that the sufficient condition that the driving strength exceeds the damping constant is necessary for the system with weak coupling to possess localized solutions.

Analytical solutions of the discrete nonlinear Schrödinger equation in arrays of optical fibers

Abstract We find the analytical solutions of the discrete nonlinear Schrodinger equation which models arrays of optical fibers according to the fact that the signal which propagate in the fiber under consideration of the array can be prone to the lateness or accelerating effects of the neighbouring fibers. Firstly, we give the recurrence relation which governs the number of fiber “ n ”and secondly, we apply the projective Riccati method to achieve the resolution.

On the collapse arresting effects of discreteness

We examine the effects of discreteness on a prototypical example of a collapse exhibiting partial differential equation (PDE). As our benchmark example, we select the discrete nonlinear Schrödinger (DNLS) equation. We provide a number of physical settings where issues of the interplay of collapse and discreteness may arise and focus on the quintic, one-dimensional DNLS. We justify that collapse in the sense of continuum limit (i.e., of the L ∞ norm becoming infinite) cannot occur in the discrete setting. We support our qualitative arguments both with numerical simulations as well as with an analysis of a quasi-continuum, pseudo-differential approximation to the discrete model. Global well-posedness is proved for the latter problem in H s , for s > 1 / 2 . While the collapse arresting nature of discreteness can be immediately realized, our estimates elucidate the “approach” towards the collapse-bearing continuum limit and the mechanism through which focusing arises in the latter.

Statistical Approach to Modulational Instability in Nonlinear Discrete Systems

We use a statistical approach to investigate the modulational instability (Benjamin-Feir instability) in several nonlinear discrete systems: the discrete nonlinear Schrodinger (NLS) equation, the Ablowitz-Ladik equation, and the discrete deformable NLS equation. We derive a kinetic equation for the two-point correlation function and use a Wigner-Moyal transformation to write it in a mixed space-wave-number representation. We perform a linear stability analysis of the resulting equation and discuss the obtained integral stability condition using several forms of the initial unperturbed spectrum (Lorentzian and δ-spectrum). We compare the results with the continuum limit (the NLS equation) and with previous results.

Jacobian elliptic function method for nonlinear differential-difference equations

Abstract An algorithm is devised to derive exact travelling wave solutions of differential-difference equations by means of Jacobian elliptic function. For illustration, we apply this method to solve the discrete nonlinear Schrodinger equation, the discretized mKdV lattice equation and the Hybrid lattice equation. Some explicit and exact travelling wave solutions such as Jacobian doubly periodic solutions, kink-type solitary wave solutions are constructed.

Breather solutions in the diffraction managed NLS equation

We show the existence of localized breather solutions in an averaged version of the discrete nonlinear Schrödinger equation (NLS) with diffraction management, a system that models coupled waveguide arrays with periodic diffraction management geometries. The breather solutions are constrained extrema of the Hamiltonian of the averaged system and their existence is shown by a discrete version of the concentration-compactness principle. The main assumptions are that the averaged diffraction is sufficiently small (compared to strength of the nonlinearity) and that the sign of the nonlinearity corresponds to the focusing case. An interesting feature of the problem is that the nonlinear interaction between neighboring lattice sites can be large and is of infinite range. On the other hand, the interaction decays rapidly at sufficiently large distances, and this plays an important role in the proof. The results also apply to higher dimensional lattices, and to the discrete NLS equation.

Exact discrete soliton solutions of quintic discrete nonlinear Schrödinger equation

By using the extended hyperbolic function approach, we have studied a quintic discrete nonlinear Schrodinger equation and obtained new exact localized solutions, including the discrete bright soliton solution, dark soliton solution, alternating phase bright soliton solution and alternating phase dark soliton solution, if a special constraint is imposed on the coefficients of the equation.

On localized solutions of discrete nonlinear Schrödinger equation: An exact result

Local and global existence of localized solutions of a discrete nonlinear Schrödinger (DNLS) equation, with arbitrary on-site nonlinearity, is proved. In particular, it is shown that an initially localized excitation persists localized during infinite time. Moreover, if initial localization is stronger than | n | − d with any power d, it maintains itself as such during infinite time. The results are generalized to various types of inter-side and saturable nonlinearities, to lattices with long range interactions, as well as DNLS with dissipation.

Bifurcations of travelling wave solutions in the discrete NLS equations

We study discrete nonlinear Schrödinger (NLS) equations, which include the cubic NLS lattice with on-site interactions and the integrable Ablowitz–Ladik lattice. Standing wave solutions are known to exist in the discrete NLS equations outside of the finite spectral band. We study travelling wave solutions which have nonlinear resonances with unbounded linear spectrum. By using center manifold and normal form reductions, we show that a continuous NLS equation with the third-order derivative term is a canonical normal form for the discrete NLS equation near the zero-dispersion limit. Bifurcations of travelling wave solutions near the zero-dispersion limit are analyzed in the framework of the third-order derivative NLS equation. We show that there exists a continuous two-parameter family of single-humped travelling wave solutions in the third-order derivative NLS equation, when it is derived from the integrable Ablowitz–Ladik lattice. On the contrary, there are no single-humped solutions in the third-order derivative NLS equation, when it is derived from the cubic NLS equation with on-site interactions. Nevertheless, we show that there exists an infinite discrete set of one-parameter families of double-humped travelling wave solutions in the latter case. Our results are valid in the neighborhood of the zero-dispersion point on the two-parameter plane of travelling wave solutions.

Path integral formulation of the tunneling dynamics of a superfluid Fermi gas in an optical potential

To describe the tunneling dynamics of a stack of two-dimensional fermionic superfluids in an optical potential, we derive an effective action functional from a path integral treatment. This effective action leads, in the saddle point approximation, to equations of motion for the density and the phase of the superfluid Fermi gas in each layer. In the strong coupling limit (where bosonic molecules are formed) these equations reduce to a discrete nonlinear Schrodinger equation, where the molecular tunneling amplitude is reduced for large binding energies. In the weak coupling (BCS) regime, we study the evolution of the stacked superfluids and derive an approximate analytical expression for the Josephson oscillation frequency in an external harmonic potential. Both in the weak and intermediate coupling regimes the detection of the Josephson oscillations described by our path integral treatment constitutes experimental evidence for the fermionic superfluid regime.

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.

Gap and out-gap breathers in a binary modulated discrete nonlinear Schr�dinger model

We consider a modulated discrete nonlinear Schrodinger (DNLS) model with alternating on-site potential, having a linear spectrum with two branches separated by a ‘forbidden’ gap. Nonlinear localized time-periodic solutions with frequencies in the gap and near the gap -- discrete gap and out-gap breathers (DGBs and DOGBs) -- are investigated. Their linear stability is studied varying the system parameters from the continuous to the anti-continuous limit, and different types of oscillatory and real instabilities are revealed. It is shown, that generally DGBs in infinite modulated DNLS chains with hard (soft) nonlinearity do not possess any oscillatory instabilities for breather frequencies in the lower (upper) half of the gap. Regimes of ‘exchange of stability’ between symmetric and antisymmetric DGBs are observed, where an increased breather mobility is expected. The transformation from DGBs to DOGBs when the breather frequency enters the linear spectrum is studied, and the general bifurcation picture for DOGBs with tails of different wave numbers is described. Close to the anti-continuous limit, the localized linear eigenmodes and their corresponding eigenfrequencies are calculated analytically for several gap/out-gap breather configurations, yielding explicit proof of their linear stability or instability close to this limit.

Higher-order lattice diffraction: solitons in the discrete NLS equation with next-nearest-neighbor interactions

We introduce a general model of a one-dimensional dynamical lattice with the on-site cubic nonlinearity and both nearest-neighbor (NN) and next-nearest-neighbor (NNN) linear interactions between lattice sites (i.e., a generalized discrete nonlinear Schrödinger equation). Unlike some previously considered cases, our model includes a complex coefficient of the NNN interaction, which applies, e.g., to optical waveguide arrays, where fields in adjacent cores may be phase shifted. The application to optical arrays is especially important in the case in which the effective lattice diffraction coefficient generated by the NN coupling is close to zero, which may be achieved by means of the so-called diffraction management. Three types of fundamental solitons are considered: site-centered and intersite-centered ones, and twisted localized modes (TLMs). It is found that, with the increase of the imaginary part of the NNN coupling constant, site-centered solitons lose their stability, but then regain it. The instability region disappears if the real part of the NNN coupling constant is negative and sufficiently large. If the site-centered soliton is unstable, it rearranges itself into a quasi-periodic (in time) breathing soliton. Intersite-centered solitons cannot be fully stabilized by the NNN interactions. TLM solitons are stable in a limited parametric region, then they become unstable, and eventually disappear. Direct simulations of the evolution of the intersite-centered solitons and unstable TLMs show that the instability reshapes them into site-centered solitons with intrinsic vibrations (breathers).

Dynamics of a period-3 pattern-loaded Bose-Einstein condensate in an optical lattice

We discuss the dynamics of a Bose-Einstein condensate initially loaded into every third site of an optical lattice using a description based upon the discrete nonlinear Schrodinger equation. An analytic solution is developed for the case of a periodic initial condition and is compared with numerical simulations for more general initial configurations. We show that mean field effects in this system can cause macroscopic quantum self-trapping, a phenomenon already predicted for double well systems. In the presence of a uniform external potential, the atoms exhibit generalized Bloch oscillations which can be interpreted in terms of the interference of three different Bloch states. We also discuss how the momentum distribution of the system can be used as experimental signature of the macroscopic self trapping effect.

Localization in physical systems described by discrete nonlinear Schrodinger-type equations.

Following a short introduction on localized modes in a model system, namely the discrete nonlinear Schrodinger equation, we present explicit results pertaining to three different physical systems described by similar equations. The applications range from the Raman scattering spectra of a complex electronic material through intrinsic localized vibrational modes, to the manifestation of an abrupt and irreversible delocalizing transition of Bose-Einstein condensates trapped in two-dimensional optical lattices, and to the instabilities of localized modes in coupled arrays of optical waveguides.

Homoclinic Tubes in Discrete Nonlinear Schrödinger Equation under Hamiltonian Perturbations

In this paper, we study the discrete cubic nonlinear Schrodinger lattice under Hamiltonian perturbations. First we develop a complete isospectral theory relevant to the hyperbolic structures of the lattice without perturbations. In particular, Backlund–Darboux transformations are utilized to generate heteroclinic orbits and Melnikov vectors. Then we give coordinate-expressions for persistent invariant manifolds and Fenichel fibers for the perturbed lattice. Finally based upon the above machinery, existence of codimension 2 transversal homoclinic tubes is established through a Melnikov type calculation and an implicit function argument. We also discuss symbolic dynamics of invariant tubes each of which consists of a doubly infinite sequence of curve segments when the lattice is four dimensional. Structures inside the asymptotic manifolds of the transversal homoclinic tubes are studied, special orbits, in particular homoclinic orbits and heteroclinic orbits when the lattice is four dimensional, are studied.

Spatial optical solitons in waveguide arrays

We overview theoretical and experimental results on spatial optical solitons excited in arrays of nonlinear waveguides. First, we briefly summarize the basic properties of the discrete nonlinear Schrodinger (NLS) equation frequently employed to study spatially localized modes in arrays, the so-called discrete solitons. Then, we introduce an improved analytical model that describes a periodic structure of thin-film nonlinear waveguides embedded into an otherwise linear dielectric medium. Such a model of waveguide arrays goes beyond the discrete NLS equation and allows studying many new features of the nonlinear dynamics in arrays, including the complete bandgap spectrum, modulational instability of extended modes, different types of gap solitons, the mode oscillatory instability, the instability-induced soliton dynamics, etc. Additionally, we summarize the recent experimental results on the generation and steering of spatial solitons and diffraction management in waveguide arrays. We also demonstrate that many effects associated with the dynamics of discrete gap solitons can be observed in a binary waveguide array. Finally, we discuss the important concept of two-dimensional (2-D) networks of nonlinear waveguides, not yet verified experimentally, which provides a roadmap for the future developments of this field. In particular, 2-D networks of nonlinear waveguides may allow a possibility of realizing useful functional operations with discrete solitons such as blocking, routing, and time gating.

Ablowitz Ladik hierarchy and two-component Toda lattice hierarchy

It is shown in the language of free fermions and the τ functions that the Ablowitz–Ladik (AL) hierarchy and the relativistic Toda lattice hierarchy arise from the A(1)1 reduction of the two-component Toda lattice hierarchy. This result gives us a simple explanation of the reason why the AL hierarchy includes important integrable systems such as the discrete nonlinear Schrodinger equation, the relativistic Toda lattice equation, the Toda field equation, the Davey–Stewartson equation, etc. We also propose a new Backlund transformation (or a discretized time system) for the AL hierarchy.

Localization-delocalization in aperiodic systems

Spectral and dynamical properties of electrons, phonons, electromagnetic waves, and nonlinear coherent excitations in one-dimensional modulated structures with long-range correlations are investigated from a theoretical point of view. First a proof of singular continuous electron spectrum for the tight-binding Schrodinger equation with an on-site potential, which, in analogy with a random potential, has an absolutely continuous correlation measure, is given. The critical behavior of such a localization phenomenon manifests in anomalous diffusion for the time-evolution of electronic wave packets. Spectral characterization of elastic vibrations in aperiodically ordered diatomic chains in the harmonic approximation is achieved through a dynamical system induced by the trace maps of renormalized transfer matrices. These results suggest that the zero Lebesgue measure Cantor-set spectrum (without eigenvalues) of the Fibonacci model for a quasicrystal is generic for deterministic aperiodic superlattices, for which the modulations take values via substitution rules on finite sets, independent of the correlation measure. Secondly, a method to synthesize and analyze discrete systems with prescribed long-range correlated disorder based on the conditional probability function of an additive Markov chain is effectively implemented. Complex gratings (artificial solids) that simultaneously display given characteristics of quasiperiodic crystals and amorphous solids on the Fraunhofer diffraction are designated. A mobility edge within second order perturbation theory of the tight-binding Schrodinger equation with a correlated disorder in the dichotomic potential realizes the success of the method in designing window filters with specific spectral components. The phenomenon of self-localization in lattice dynamical systems is a subject of interest in various physical disciplines. Lattice solitons are studied using the discrete nonlinear Schrodinger equation with on-site potential, modeling coherent structures in, for example, photonic crystals. The instability-induced dynamics of the localized gap soliton is found to thermalize according to the Gibbsian equilibrium distribution, while the spontaneous formation of persisting intrinsic localized modes from the extended out-gap soliton reveals a phase transition of the solution.

Standing wave instabilities, breather formation and thermalization in a Hamiltonian anharmonic lattice

Modulational instability of travelling plane waves is often considered as the first step in the formation of intrinsically localized modes (discrete breathers) in anharmonic lattices. Here, we consider an alternative mechanism for breather formation, originating in oscillatory instabilities of spatially periodic or quasiperiodic nonlinear standing waves (SWs). These SWs are constructed for Klein-Gordon or Discrete Nonlinear Schrodinger lattices as exact time periodic and time reversible multibreather solutions from the limit of uncoupled oscillators, and merge into harmonic SWs in the small-amplitude limit. Approaching the linear limit, all SWs with nontrivial wave vectors (0 < Q < π) become unstable through oscillatory instabilities, persisting for arbitrarily small amplitudes in infinite lattices. The dynamics resulting from these instabilities is found to be qualitatively different for wave vectors smaller than or larger than π/2, respectively. In one regime persisting breathers are found, while in the other regime the system thermalizes.