Discrete Nonlinear Schrodinger Research Articles

Exponential Times in the One-Dimensional Gross–Pitaevskii Equation with Multiple Well Potential

We consider the Gross-Pitaevskii equation in 1 space dimension with a N-well trapping potential. We prove, in the semiclassical limit, that the finite dimensional eigenspace associated to the lowest N eigenvalues of the linear operator is slightly deformed by the nonlinear term into an almost invariant manifold \({\mathcal{M}}\) . Precisely, one has that solutions starting on \({\mathcal{M}}\) , or close to it, will remain close to \({\mathcal{M}}\) for times exponentially long with the inverse of the size of the nonlinearity. As heuristically expected the effective equation on \({\mathcal M}\) is a perturbation of a discrete nonlinear Schrodinger equation. We deduce that when the size of the nonlinearity is large enough then tunneling among the wells essentially disappears: that is for almost all solutions starting close to \({\mathcal{M}}\) their restriction to each of the wells has norm approximatively constant over the considered time scale. In the particular case of a double well potential we give a more precise result showing persistence or destruction of the beating motions over exponentially long times. The proof is based on canonical perturbation theory; surprisingly enough, due to the Gauge invariance of the system, no non-resonance condition is required.

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We consider the Gross-Pitaevskii equation in 1 space dimension with a N-well trapping potential. We prove, in the semiclassical limit, that the finite dimensional eigenspace associated to the lowest N eigenvalues of the linear operator is slightly deformed by the nonlinear term into an almost invariant manifold \({\mathcal{M}}\) . Precisely, one has that solutions starting on \({\mathcal{M}}\) , or close to it, will remain close to \({\mathcal{M}}\) for times exponentially long with the inverse of the size of the nonlinearity. As heuristically expected the effective equation on \({\mathcal M}\) is a perturbation of a discrete nonlinear Schrodinger equation. We deduce that when the size of the nonlinearity is large enough then tunneling among the wells essentially disappears: that is for almost all solutions starting close to \({\mathcal{M}}\) their restriction to each of the wells has norm approximatively constant over the considered time scale. In the particular case of a double well potential we give a more precise result showing persistence or destruction of the beating motions over exponentially long times. The proof is based on canonical perturbation theory; surprisingly enough, due to the Gauge invariance of the system, no non-resonance condition is required.

One-parameter localized traveling waves in nonlinear Schrödinger lattices

We address the existence of traveling single-humped localized solutions in the spatially discrete nonlinear Schrödinger (NLS) equation. A mathematical technique is developed for analysis of persistence of these solutions from a certain limit in which the dispersion relation of linear waves contains a triple zero. The technique is based on using the Implicit Function Theorem for solution of an appropriate differential advance–delay equation in exponentially weighted spaces. The resulting Melnikov calculation relies on a number of assumptions on the spectrum of the linearization around the pulse, which are checked numerically. We apply the technique to the so-called Salerno model and the translationally invariant discrete NLS equation with a cubic nonlinearity. We show that the traveling solutions terminate in the Salerno model whereas they generally persist in the translationally invariant NLS lattice as a one-parameter family of solutions. These results are found to be in a close correspondence with numerical approximations of traveling solutions with zero radiation tails. Analysis of persistence also predicts the spectral stability of the one-parameter family of traveling solutions under time evolution of the discrete NLS equation.

Elementary Vortex Filament Model of the Discrete Nonlinear Schrödinger Equation

The discretization of a vortex filament is studied. A discrete vortex model and a discrete version of the local induction approximation technique (LIA), which guarantees that the vortex filament obeys the discrete nonlinear Schrodinger equation, is proposed.

A simple but efficient approach for studying on nonlinear differential-difference equations

In this paper, we present a new approach for constructing exact solutions to nonlinear differential-difference equations (NLDDEs). By applying the new method, we have studied the saturable discrete nonlinear Schrodinger equation (SDNLSE) and obtained a number of new exact localized solutions, including discrete bright soliton solution, dark soliton solution, bright and dark soliton solution, alternating phase bright soliton solution, alternating phase dark soliton solution and alternating phase bright and dark soliton solution, provided that a special relation is bound on the coefficients of the equation among the solutions obtained.

Exact stationary solutions for the translationally invariant discrete nonlinear Schrödinger equations

From a wide class of translationally invariant discrete nonlinear Schrödinger (DNLS) equations, we extract a two-parameter subclass corresponding to Kerr nonlinearity for which any stationary solution can be derived recurrently from a quadratic equation. This subclass, which incorporates the integrable (Ablowitz–Ladik) lattice as a special case, admits exact stationary solutions that are derived in terms of the Jacobi elliptic functions. Exact moving solutions for the discrete equations are also obtained. In the continuum limit, the constructed stationary solutions reduce to the exact moving solutions to the continuum NLS equation with Kerr nonlinearity. Numerical results are also presented for the special case of localized solutions, including sech (pulse, or bright soliton), tanh (kink, or dark soliton) and 1/tanh (called here inverted kink) profiles. For these solutions, we discuss their linearization spectra and their mobility. Particularly, we demonstrate that discrete dark solitons are dynamically stable for a wide range of lattice spacings, contrary to what is the case for their standard DNLS counterparts. Furthermore, the bright and dark solitons in the non-integrable, translationally invariant lattices can propagate at slow speed without any noticeable radiation.

The existence of a global attractor for the discrete nonlinear Schrödinger equation. II. Compactness without tail estimates in $\mathbb{Z}^N$, $N\geq1$, lattices

Recent studies of lattice dynamics, with respect to the existence of global compact attractors for certain discrete evolution equations, are based on the derivation of ‘tail estimates of the solution’. We show that, due to the specific nature of the discrete nonlinear Schrodinger (DNLS) system which gives rise to a simple energy equation, the method developed by J. M. Ball is applicable, and provides an alternative proof on the existence of the compactness of the attractor for the weakly damped and driven DNLS equation considered in $\mathbb{Z}^N$ , $N\geq1$ , lattices, without the usage of tail estimates. The approach covers various DNLS-type equations of physical significance.

Modulational instabilities in acetanilide taking into account both the N–H and the C=O vibrational self-trappings

A model of crystalline acetanilide, ACN accounting for the C=O and N–H vibrational self-trappings is presented. We develop a fully discrete version of ACN. We show that ACN can be described by a set of two coupled discrete nonlinear Schrödinger (DNLS) equations. Modulational instabilities (MI) are studied both theoretically and numerically. Dispersion laws for the wavenumbers and frequencies of the linear modulation waves are determined. We also derived the criterion for the existence of MI. Numerical simulations are carried out for a variety of selected wave amplitudes in the unstable zone. It is shown that instabilities grow as the wavenumbers and amplitudes of the modulated waves increase. MI grow faster in the N–H mode than in the C=O mode. Temporal evolution of the density probabilities of the vibrational excitons are obtained by the numerical integration of the coupled DNLS equations governing the ACN molecule. These investigations confirm the generation of localized modes by the phenomenon of MI and the predominance of the N–H vibrational mode in the MI process of the ACN.

PROPERTIES OF ENERGY SPECTRA OF MOLECULAR CRYSTALS INVESTIGATED BY NONLINEAR THEORY

We calculate the quantum energy spectra of molecular crystals, such as acetanilide, by using discrete nonlinear Schrodinger equation, containing various interactions, appropriate to the systems. The energy spectra consist of many energy bands, in each energy band there are a lot of energy levels including some higher excited states. The result of energy spectrum is basically consistent with experimental values obtained by infrared absorption and Raman scattering in acetanilide and can also explain some experimental results obtained by Careri et al. Finally, we further discuss the influences of variously characteristic parameters on the energy spectra of the systems.

On a class of spatial discretizations of equations of the nonlinear Schrödinger type

We demonstrate the systematic derivation of a class of discretizations of nonlinear Schrödinger (NLS) equations for general polynomial nonlinearity whose stationary solutions can be found from a reduced two-point algebraic condition. We then focus on the cubic problem and illustrate how our class of models compares with the well-known discretizations such as the standard discrete NLS equation, or the integrable variant thereof. We also discuss the conservation laws of the derived generalizations of the cubic case, such as the lattice momentum or mass and the connection with their corresponding continuum siblings.

Transition to chaos in discrete nonlinear Schrödinger equation with long-range interaction

Discrete nonlinear Schrödinger (DNLS) equation describes a chain of oscillators with nearest-neighbor interactions and a specific nonlinear term. We consider its modification with long-range interaction through a potential proportional to 1 / l 1 + α with fractional α < 2 and l as a distance between oscillators. This model is called α DNLS. It exhibits competition between the nonlinearity and a level of correlation between interacting far-distanced oscillators, that is defined by the value of α . We consider transition to chaos in this system as a function of α and nonlinearity. It is shown that decreasing of α with respect to nonlinearity stabilize the system. Connection of the model to the fractional generalization of the NLS (called FNLS) in the long-wave approximation is also discussed and some of the results obtained for α DNLS can be correspondingly extended to the FNLS.

Translationally invariant nonlinear Schrödinger lattices

The persistence of stationary and travelling single-humped localized solutions in the spatial discretizations of the nonlinear Schrödinger (NLS) equation is addressed. The discrete NLS equation with the most general cubic polynomial function is considered. Constraints on the nonlinear function are found from the condition that the second-order difference equation for stationary solutions can be reduced to the first-order difference map. The discrete NLS equation with such an exceptional nonlinear function is shown to have a conserved momentum but admits no standard Hamiltonian structure. It is proved that the reduction to the first-order difference map gives a sufficient condition for existence of translationally invariant single-humped stationary solutions. Another constraint on the nonlinear function is found from the condition that the differential advance–delay equation for travelling solutions admits a reduction to an integrable normal form given by a third-order differential equation. This reduction gives a necessary condition for existence of single-humped travelling solutions. The nonlinear function which admits both reductions defines a four-parameter family of discrete NLS equations which generalizes the integrable Ablowitz–Ladik lattice. Particular travelling solutions of this family of discrete NLS equations are written explicitly.

GLOBAL EXISTENCE IN INFINITE LATTICES OF NONLINEAR OSCILLATORS: THE DISCRETE KLEIN-GORDON EQUATION

Pointing out the difference between the Discrete Nonlinear Schrodinger equation with the classical power law nonlinearity – for which solutions exist globally, independently of the sign and the degree of the nonlinearity, the size of the initial data and the dimension of the lattice – we prove either global existence or nonexistence, in time, for the Discrete Klein-Gordon equation with the same type of nonlinearity (but of “blow-up” sign), under suitable conditions on the initial data, and sometimes on the dimension of the lattice. The results consider both the conservative and the linearly damped lattice. Similarities and differences with the continuous counterparts are remarked. We also make a short comment on the existence of excitation thresholds, for forced solutions of damped and parametrically driven Klein-Gordon lattices.

Modulational instability and spatial structures of the Ablowitz–Ladik equation

Abstract Spatial structures as a result of a modulational instability are obtained in the integrable discrete nonlinear Schrodinger equation (Ablowitz–Ladik equation). Discrete slow space variables are used in a general setting and the related finite differences are constructed. Analyzing the ensuing equation, we derive the modulational instability criterion from the discrete multiple scales approach. Numerical simulations in agreement with analytical studies lead to the disintegrations of the initial modulated waves into a train of pulses.

Multi-peaked localized states of DNLS in one and two dimensions

Multi-peaked localized stationary solutions of the discrete nonlinear Schrödinger (DNLS) equation are presented in one (1D) and two (2D) dimensions. These are excited states of the discrete spectrum and correspond to multi-breather solutions. A simple, very fast, and efficient numerical method, suggested by Aubry, has been used for their calculation. The method involves no diagonalization, but just iterations of a map, starting from trivial solutions of the anti-continuous limit. Approximate analytical expressions are presented and compared with the numerical results. The linear stability of the calculated stationary states is discussed and the structure of the linear stability spectrum is analytically obtained for relatively large values of nonlinearity.

Discrete nonlinear Schrödinger equations free of the Peierls–Nabarro potential

We derive a class of discrete nonlinear Schrödinger (DNLS) equations for general polynomial nonlinearity whose stationary solutions can be found from a reduced two-point algebraic problem. It is demonstrated that the derived class of discretizations contains subclasses conserving classical norm or a modified norm and classical momentum. These equations are interesting from the physical standpoint since they support stationary discrete solitons free of the Peierls–Nabarro potential. Focusing on the cubic nonlinearity we then consider a small perturbation around stationary soliton solutions and, solving corresponding eigenvalue problem, we (i) demonstrate that solitons are stable; (ii) show that they have two additional zero-frequency modes responsible for their effective translational invariance; (iii) derive semi-analytical solutions for discrete solitons moving at slow speed. To highlight the unusual properties of solitons in the new discrete models we compare them with that of the classical DNLS equation giving several numerical examples.

A method for constructing discrete exact solutions and application to quintic discrete nonlinear Schrödinger equation☆

Abstract In this paper, we present a method to solve nonlinear difference differential equation(s). We applied this method to quintic discrete nonlinear Schrodinger equation to illustrate this method. Many exact solutions are obtained with the help of Maple including soliton solutions presented by hyperbolic functions sinh and cosh, periodic solutions presented by sin and cos and rational solutions. This method can also be used to other nonlinear difference-differential equation(s).

Multistable solitons in the cubic–quintic discrete nonlinear Schrödinger equation

We analyze the existence and stability of localized solutions in the one-dimensional discrete nonlinear Schrödinger (DNLS) equation with a combination of competing self-focusing cubic and defocusing quintic onsite nonlinearities. We produce a stability diagram for different families of soliton solutions that suggests the (co)existence of infinitely many branches of stable localized solutions. Bifurcations that occur with an increase in the coupling constant are studied in a numerical form. A variational approximation is developed for accurate prediction of the most fundamental and next-order solitons, together with their bifurcations. Salient properties of the model, which distinguish it from the well-known cubic DNLS equation, are the existence of two different types of symmetric solitons and stable asymmetric soliton solutions that are found in narrow regions of the parameter space. The asymmetric solutions appear from and disappear back into the symmetric ones via loops of forward and backward pitchfork bifurcations.

Discrete light localization in one-dimensional nonlinear lattices with arbitrary nonlocality

We model discrete spatial solitons in a periodic nonlinear medium encompassing any degree of transverse nonlocality. Making a convenient reference to a widely used material--nematic liquid crystals--we derive a form of the discrete nonlinear Schrödinger equation and find a family of discrete solitons. Such self-localized solutions in optical lattices can exist with an arbitrary degree of imprinted chirp and have breathing character. We verify numerically that both local and nonlocal discrete light propagation and solitons can be observed in liquid crystalline arrays.

Standing waves in a non-linear 1D lattice: Floquet multipliers, Krein signatures, and stability

We construct a class of exact commensurate and incommensurate standing wave (SW) solutions in a piecewise smooth analogue of the discrete non-linear Schrödinger (DNLS) model and present their linear stability analysis. In the case of the commensurate SW solutions the analysis reduces to the eigenvalue problem of a transfer matrix depending parametrically on the eigenfrequency. The spectrum of eigenfrequencies and the corresponding eigenmodes can thereby be determined exactly. The spatial periodicity of a commensurate SW implies that the eigenmodes are of the Bloch form, characterised by an even number of Floquet multipliers. The spectrum is made up of bands that, in general, include a number of transition points corresponding to changes in the disposition of the Floquet multipliers. The latter characterise the different band segments. An alternative characterisation of the segments is in terms of the Krein signatures associated with the eigenfrequencies. When one or more parameters characterising the SW solution is made to vary, one occasionally encounters collisions between the band-edges or the intra-band transition points and, depending on the the Krein signatures of the colliding bands or segments, the spectrum may stretch out in the complex plane, leading to the onset of instability. We elucidate the correlation between the disposition of Floquet multipliers and the Krein signatures, presenting two specific examples where the SW possesses a definite window of stability, as distinct from the SW’s obtained close to the anti-continuous and linear limits of the DNLS model.