Discrete Nonlinear Schroedinger Equation Research Articles

Ground states in spatially discrete non-linear Schrödinger models

In the seminal work (Weinstein 1999 Nonlinearity 12 673), Weinstein considered the question of the ground states for discrete Schrödinger equations with power law nonlinearities, posed on . More specifically, he constructed the so-called normalised waves, by minimising the Hamiltonian functional, for fixed power P (i.e. l 2 mass). This type of variational method allows one to claim, in a straightforward manner, set stability for such waves. In this work, we revisit these questions and build upon Weinstein’s work, as well as the innovative variational methods introduced for this problem in (Laedke et al 1994 Phys. Rev. Lett. 73 1055 and Laedke et al 1996 Phys. Rev. E 54 4299) in several directions. First, for the normalised waves, we show that they are in fact spectrally stable as solutions of the corresponding discrete nonlinear Schroedinger equation (NLS) evolution equation. Next, we construct the so-called homogeneous waves, by using a different constrained optimisation problem. Importantly, this construction works for all values of the parameters, e.g. l 2 supercritical problems. We establish a rigorous criterion for stability, which decides the stability on the homogeneous waves, based on the classical Grillakis–Shatah–Strauss/Vakhitov–Kolokolov (GSS/VK) quantity . In addition, we provide some symmetry results for the solitons. Finally, we complement our results with numerical computations, which showcase the full agreement between the conclusion from the GSS/VK criterion vis-á-vis with the linearised problem. In particular, one observes that it is possible for the stability of the wave to change as the spectral parameter ω varies, in contrast with the corresponding continuous NLS model.

Modeling reservoir computing with the discrete nonlinear Schrödinger equation

We formulate, using the discrete nonlinear Schroedinger equation (DNLS), a general approach to encode and process information based on reservoir computing. Reservoir computing is a promising avenue for realizing neuromorphic computing devices. In such computing systems, training is performed only at the output level, by adjusting the output from the reservoir with respect to a target signal. In our formulation, the reservoir can be an arbitrary physical system, driven out of thermal equilibrium by an external driving. The DNLS is a general oscillator model with broad application in physics and we argue that our approach is completely general and does not depend on the physical realisation of the reservoir. The driving, which encodes the object to be recognised, acts as a thermodynamical force, one for each node in the reservoir. Currents associated to these thermodynamical forces in turn encode the output signal from the reservoir. As an example, we consider numerically the problem of supervised learning for pattern recognition, using as reservoir a network of nonlinear oscillators.

Bloch spin waves and emergent structure in protein folding with HIV envelope glycoprotein as an example.

We inquire how structure emerges during the process of protein folding. For this we scrutinize collective many-atom motions during all-atom molecular dynamics simulations. We introduce, develop, and employ various topological techniques, in combination with analytic tools that we deduce from the concept of integrable models and structure of discrete nonlinear Schrödinger equation. The example we consider is an α-helical subunit of the HIV envelope glycoprotein gp41. The helical structure is stable when the subunit is part of the biological oligomer. But in isolation, the helix becomes unstable, and the monomer starts deforming. We follow the process computationally. We interpret the evolving structure both in terms of a backbone based Heisenberg spin chain and in terms of a side chain based XY spin chain. We find that in both cases the formation of protein supersecondary structure is akin the formation of a topological Bloch domain wall along a spin chain. During the process we identify three individual Bloch walls and we show that each of them can be modelled with a precision of tenths to several angstroms in terms of a soliton solution to a discrete nonlinear Schrödinger equation.

Solitary wave dynamics in transversely and longitudinally modulated nonlocal media

The propagation of optical beams in a non-local nonlinear medium which is modulated both transversely and longitudinally is investigated. The employed model is a generic discrete nonlinear Schroedinger (DNLS) equation, with a direct modulation of the degree of nonlocality. The focus is on media such as nematic liquid crystals. A variational method is utilized for the systematic investigation of the dependence of soliton dynamics on both its initial characteristics and the configuration parameters. The longitudinal modulation is shown to result in dynamical features of soliton propagation that have no counterpart in longitudinally homogeneous media, suggesting additional soliton propagation control capabilities.

Nonequilibrium discrete nonlinear Schrödinger equation

We study nonequilibrium steady states of the one-dimensional discrete nonlinear Schrödinger equation. This system can be regarded as a minimal model for the stationary transport of bosonic particles such as photons in layered media or cold atoms in deep optical traps. Due to the presence of two conserved quantities, namely, energy and norm (or number of particles), the model displays coupled transport in the sense of linear irreversible thermodynamics. Monte Carlo thermostats are implemented to impose a given temperature and chemical potential at the chain ends. As a result, we find that the Onsager coefficients are finite in the thermodynamic limit, i.e., transport is normal. Depending on the position in the parameter space, the "Seebeck coefficient" may be either positive or negative. For large differences between the thermostat parameters, density and temperature profiles may display an unusual nonmonotonic shape. This is due to the strong dependence of the Onsager coefficients on the state variables.

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 for the Gross-Pitaevskii equation with two-dimensional periodic potentials and for the discrete nonlinear Schrödinger 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 the way to the band edges.

Scaling properties of weak chaos in nonlinear disordered lattices

We study the discrete nonlinear Schrödinger equation with a random potential in one dimension. It is characterized by the length, the strength of the random potential, and the field density that determines the effect of nonlinearity. Following the time evolution of the field and calculating the largest Lyapunov exponent, the probability of the system to be regular is established numerically and found to be a scaling function of the parameters. This property is used to calculate the asymptotic properties of the system in regimes beyond our computational power.

Universal character of the discrete nonlinear Schrödinger equation

We address the universal applicability of the discrete nonlinear Schroedinger equation. By employing an original but general top-down-bottom-up procedure based on symmetry analysis to the case of optical lattices, we derive the most widely applicable and simplest possible model, revealing that the discrete nonlinear Schroedinger equation is 'universally' fit to describe light propagation even in discrete tensorial nonlinear systems and in the presence of nonparaxial and vectorial effects.

Localization and trapping of light in one- and two-dimensional nonlinear periodic structures

Novel photonic crystals have been proposed and built where the phenomenon of light localization occurs. This localization takes many forms, including discrete and continuous gap solitons, trapping via defect modes, localization due to randomness, just to name some. In this paper we discuss pulse dynamics in nonlinear one-dimensional fiber Bragg gratings and two-dimensional periodic waveguides. Here, gap solitons propagate and are trapped into defect via resonant interactions. A third case we consider is that of light propagation in nonlinear fiber arrays where we add randomness that model imperfections in the separation between fibers. The model which describes the dynamics is the discrete nonlinear Schroedinger equation with random coupling coefficients. The question we address is whether light localization as a balance of discrete diffraction and nonlinearity is robust when random effects are present.

Ground-state properties of small-size nonlinear dynamical lattices

We investigate the ground state of a system of interacting particles in small nonlinear lattices with M >or=3 sites, using as a prototypical example the discrete nonlinear Schrödinger equation that has been recently used extensively in the contexts of nonlinear optics of waveguide arrays and Bose-Einstein condensates in optical lattices. We find that, in the presence of attractive interactions, the dynamical scenario relevant to the ground-state and the lowest-energy modes of such few-site nonlinear lattices reveals a variety of nontrivial features that are absent in the large/infinite lattice limits: the single-pulse solution and the uniform solution are found to coexist in a finite range of the lattice intersite coupling where, depending on the latter, one of them represents the ground state; in addition, the single-pulse mode does not even exist beyond a critical parametric threshold. Finally, the onset of the ground-state (modulational) instability appears to be intimately connected with a nonstandard ("double transcritical") type of bifurcation that, to the best of our knowledge, has not been reported previously in other physical systems.

Ground-state selection as a four-wave-mixing process

We study nonlinear relaxation of the excited state in a system with two levels and a continuum. The process is first simulated in a one-dimensional waveguide array described by the discrete nonlinear Schroedinger equation. The results are interpreted in terms of degenerate four-wave mixing between the eigenmodes and diffraction properties of the array. We also show analytically that the role of the continuum can be played by a third bound state, with linear loss that replaces the diffraction in the continuum. This model enables derivation of the energy transfer rate and other parameters of the process.

Simple statistical explanation for the localization of energy in nonlinear lattices with two conserved quantities.

The localization of energy in the discrete nonlinear Schrödinger equation is explained with statistical methods. The partition function and the entropy of the system are computed for low-amplitude initial conditions. Detailed predictions for the long-time solution are derived. Localized high-amplitude excitations absorb a surplus of energy when they emerge as a by-product of the production of entropy in the small fluctuations. The thermodynamic interpretation of this process applies to many dynamical systems with two conserved quantities.

Array of Bose-Einstein condensates under time-periodic Feshbach-resonance management

The dynamics of a discrete soliton in an array of Bose-Einstein condensates under the action of a periodically time-modulated atomic scattering length (``Feshbach-resonance management, FRM'') is investigated. The cases of both slow and rapid modulation, in comparison with the tunneling frequency, are considered. We employ a discrete variational approach for the analysis of the system. The existence of nonlinear resonances and chaos is predicted at special values of the driving frequency. Soliton splitting is observed in numerical simulations. In the case of the rapid modulation, we derive an averaged equation, which is a generalized discrete nonlinear Schroedinger equation, including higher-order effective nonlinearities and intersite nonlinear interactions. Thus the predicted discrete FRM solitons are a direct matter-wave analog of recently investigated discrete diffraction-managed optical solitons.

Study of some approximation schemes in the spin-boson problem

Some approximation schemes used in the description of the evolution of the spin-boson system are studied through numerical and analytic methods. Among the procedures investigated are semiclassical approximations and the memory function approach. An infinitely large number of semiclassical approximations are discussed. Their two extreme limits are shown to be characterized, respectively, by effective energy mismatch and effective intersite transfer. The validity of the two limits is explored by explicit numerical calculations for important regions in parameter space, and it is shown that they can provide good descriptions in the so-called adiabatic and anti-adiabatic regimes, respectively. The memory function approach, which provides an excellent approximation scheme for a certain range of parameters, is shown to be connected to other approaches such as the non-interacting blip approximation. New results are derived from the memory approach in semiclassical contexts. Comments are made on thermal effects in the spin-boson problem, the discrete non-linear Schroedinger equation, and connections to the areas of dynamic localization, and quantum control.

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.

Multiscale analysis of discrete nonlinear evolution equations

The method of multiscale analysis is constructed for dicrete systems of evolution equations for which the problem is that of the far behavior of an input boundary datum. Discrete slow space variables are introduced in a general setting and the related finite differences are constructed. The method is applied to a series of representative examples: the Toda lattice, the nonlinear Klein-Gordon chain, the Takeno system and a discrete version of the Benjamin-Bona-Mahoney equation. Among the resulting limit models we find a discrete nonlinear Schroedinger equation (with reversed space-time), a 3-wave resonant interaction system and a discrete modified Volterra model.

Coupled quasiparticle-boson systems: The semiclassical approximation and discrete nonlinear Schrödinger equation

The validity of the semiclassical approximation is studied for a system comprising one quasiparticle coupled to a boson degree of freedom. Using a two-site Holstein model as an example, it is shown that the semiclassical approximation becomes exact in a nontrivial adiabatic limit. Furthermore, in the model`s polaron regime, there exists a hierarchy of time scales that rationalizes the quantum dynamics of the Holstein model. For the single-mode case considered, the discrete nonlinear Schroedinger equation is found to be valid only in a highly limited antiadiabatic regime.