Discrete nonlinear Schrödinger equations with arbitrarily high-order nonlinearities

Thermalization in the one-dimensional Salerno model lattice

The Salerno model constitutes an intriguing interpolation between the integrable Ablowitz-Ladik (AL) model and the more standard (nonintegrable) discrete nonlinear Schrödinger (DNLS) one. The competition of local on-site nonlinearity and nonlinear dispersion governs the thermalization of this model. Here, we investigate the statistical mechanics of the Salerno one-dimensional lattice model in the nonintegrable case and illustrate the thermalization in the Gibbs regime. As the parameter interpolating between the two limits (from DNLS toward AL) is varied, the region in the space of initial energy and norm densities leading to thermalization expands. The thermalization in the non-Gibbs regime heavily depends on the finite system size; we explore this feature via direct numerical computations for different parametric regimes.

Dissipative solitons in the discrete Ginzburg-Landau equation with saturable nonlinearity

The modulational instability of nonlinear plane waves and the existence of periodic and localized dissipative solitons and waves of the discrete Ginzburg-Landau equation with saturable nonlinearity are investigated. Explicit analytic expressions for periodic solutions with a zero and a finite background are derived and their stability properties investigated by means of direct numerical simulations. We find that while discrete periodic waves and solitons on a zero background are stable under time evolution, they may become modulationally unstable on finite backgrounds. The effects of a linear ramp potential on stable localized dissipative solitons are also briefly discussed.

High-speed kinks in a generalized discrete phi4 model

We consider a generalized discrete phi4 model and demonstrate that it can support exact moving kink solutions in the form of tanh with an arbitrarily large velocity. The constructed exact moving solutions are dependent on the specific value of the propagation velocity. We demonstrate that in this class of models, given a specific velocity, the problem of finding the exact moving solution is integrable. Namely, this problem originally expressed as a three-point map can be reduced to a two-point map, from which the exact moving solutions can be derived iteratively. It was also found that these high-speed kinks can be stable and robust against perturbations introduced in the initial conditions.

Enigma of probability amplitudes in Hamiltonian formulation of integrable semidiscrete nonlinear Schrödinger systems

An attempt to find a probability-amplitude-based Hamiltonian representation of the symmetrical version of integrable semidiscrete multicomponent nonlinear Schrödinger systems is made. Thus the on-cell locality of the general point transformation in combination with the model multicomponentness is shown to contradict the concept of canonical Hamiltonian representation in terms of probability amplitudes. Nevertheless, the above concept can be realized in a slightly adjusted semidiscrete multicomponent nonlinear Schrödinger system that preserves some physically valuable solutions of the original (either symmetric or asymmetric) integrable model. The advantages of the adjusted Hamiltonian model for the analysis of real physical systems are formulated. Examples of longitudinal and lateral soliton dynamics on multichain tubular lattices subjected to uniform electric and magnetic fields are given.

Comparative study of different discretizations of the phi(4) model

We examine various recently proposed translationally invariant discretizations of the well-known phi(4) field theory. We compare and contrast the properties of their fundamental solutions including the nature of their kink-type solitary waves and the spectral properties of the linearization around such waves. We study these features as a function of the lattice spacing h , as one deviates from the continuum limit of h --> 0. We then proceed to a more "stringent" comparison of the models, by discussing the scattering properties of a kink-antikink pair for the different discretizations. These collisions are well known to possess properties that quite sensitively depend on the initial speed even at the continuum limit. We examine how typical model behaviors are modified in the presence (and as a function) of discreteness. One of the surprising trends that we observe is the increasing elasticity of kink collisions with deviation from the continuum limit. Another general feature is that the most inelastic kink collisions are observed in the classical discrete phi(4) model, while they are more elastic in the four studied translationally invariant models.

Solitons in strongly driven discrete nonlinear Schrödinger-type models

Discrete solitons in the Ablowitz-Ladik (AL) and discrete nonlinear Schrödinger (DNLS) equations with damping and strong rapid drive are investigated. The averaged equations have the forms of the parametric AL and DNLS equations. An additional type of parametric bright discrete soliton and cnoidal waves are found and the stability properties are analyzed. The analytical predictions of the perturbed inverse scattering transform are confirmed by the numerical simulations of the AL and DNLS equations with rapidly varying drive and damping.