Dark solitons in dynamical lattices with the cubic-quintic nonlinearity

Jackiw-Rebbi states and trivial states in interfaced binary waveguide arrays with cubic-quintic nonlinearity

We systematically investigate two types of localized states-one is the optical analog of the quantum relativistic Jackiw-Rebbi states and the other is the trivial localized state-in interfaced binary waveguide arrays in the presence of cubic-quintic nonlinearity. By using the shooting method, we can exactly calculate the profiles of these nonlinear localized states. Like in the case with Kerr nonlinearity, we demonstrate that these localized states with cubic-quintic nonlinearity also have an extraordinary property, which completely differs from many well-known nonlinear localized structures in other media. Specifically, both the peak amplitude and transverse dimension of these nonlinear localized states can increase at the same time. Apart from that, we show that high values of the saturation nonlinearity parameter can help to generate and stabilize the intense localized states during propagation, especially in the case with a negative coefficient for the cubic nonlinearity term.

Stability of optical solitons in parity-time-symmetric optical lattices with competing cubic and quintic nonlinearities

The existence and stability of optical solitons in the semi-infinite gap of parity-time (PT)-symmetric optical lattices with competing cubic and quintic nonlinearities are investigated numerically. The fundamental and dipole solitons can exist only with focusing quintic nonlinearity; however, they are always linearly unstable. With the competing effect between cubic and quintic nonlinearities, the strength of the quintic nonlinearity should be larger than a threshold for the solitons' existence when the strength of the focusing cubic nonlinearity is fixed. The stability of both fundamental and dipole solitons is studied in detail. When the strength of the focusing quintic nonlinearity is fixed, solitons can exist at the whole interval of the strength of the cubic nonlinearity, but only a small part of the fundamental solitons are stable. We also study numerically nonlinear evolution of stable and unstable PT solitons under perturbation.

Mobility of discrete multibreathers in the exciton dynamics of the Davydov model with saturable nonlinearities

We show that the state of amide-I excitations in proteins is modeled by the discrete nonlinear Schrödinger equation with saturable nonlinearities. This is done by extending the Davydov model to take into account the competition between local compression and local dilatation of the lattice, thus leading to the interplay between self-focusing and defocusing saturable nonlinearities. Site-centered (sc) mode and/or bond-centered mode like discrete multihump soliton (DMHS) solutions are found numerically and their stability is analyzed. As a result, we obtained the existence and stability diagrams for all observed types of sc DMHS solutions. We also note that the stability of sc DMHS solutions depends not only on the value of the interpeak separation but also on the number of peaks, while their counterpart having at least one intersite soliton is instable. A study of mobility is achieved and it appears that, depending on the higher-order saturable nonlinearity, DMHS-like mechanism for vibrational energy transport along the protein chain is possible.

Modulational instability of a modified Gross-Pitaevskii equation with higher-order nonlinearity

We consider the modulational instability (MI) of Bose-Einstein condensate (BEC) described by a modified Gross-Pitaevskii (GP) equation with higher-order nonlinearity both analytically and numerically. A new explicit time-dependent criterion for exciting the MI is obtained. It is shown that the higher-order term can either suppress or enhance the MI, which is interesting for control of the system instability. Importantly, we predict that with the help of the higher-order nonlinearity, the MI can also take place in a BEC with repulsively contact interactions. The analytical results are confirmed by direct numerical simulations.

Two routes to the one-dimensional discrete nonpolynomial Schrodinger equation

The Bose-Einstein condensate (BEC), confined in a combination of the cigar-shaped trap and axial optical lattice, is studied in the framework of two models described by two versions of the one-dimensional (1D) discrete nonpolynomial Schrodinger equation (NPSE). Both models are derived from the three-dimensional Gross-Pitaevskii equation (3D GPE). To produce "model 1" (which was derived in recent works), the 3D GPE is first reduced to the 1D continual NPSE, which is subsequently discretized. "Model 2," which was not considered before, is derived by first discretizing the 3D GPE, which is followed by the reduction in the dimension. The two models seem very different; in particular, model 1 is represented by a single discrete equation for the 1D wave function, while model 2 includes an additional equation for the transverse width. Nevertheless, numerical analyses show similar behaviors of fundamental unstaggered solitons in both systems, as concerns their existence region and stability limits. Both models admit the collapse of the localized modes, reproducing the fundamental property of the self-attractive BEC confined in tight traps. Thus, we conclude that the fundamental properties of discrete solitons predicted for the strongly trapped self-attracting BEC are reliable, as the two distinct models produce them in a nearly identical form. However, a difference between the models is found too, as strongly pinned (very narrow) discrete solitons, which were previously found in model 1, are not generated by model 2-in fact, in agreement with the continual 1D NPSE, which does not have such solutions either. In that respect, the newly derived model provides for a more accurate approximation for the trapped BEC.

Extreme events in discrete nonlinear lattices

We perform statistical analysis on discrete nonlinear waves generated through modulational instability in the context of the Salerno model that interpolates between the integrable Ablowitz-Ladik (AL) equation and the nonintegrable discrete nonlinear Schrödinger equation. We focus on extreme events in the form of discrete rogue or freak waves that may arise as a result of rapid coalescence of discrete breathers or other nonlinear interaction processes. We find power law dependence in the wave amplitude distribution accompanied by an enhanced probability for freak events close to the integrable limit of the equation. A characteristic peak in the extreme event probability appears that is attributed to the onset of interaction of the discrete solitons of the AL equation and the accompanied transition from the local to the global stochasticity monitored through the positive Lyapunov exponent of a nonlinear map.

Staggered and moving localized modes in dynamical lattices with the cubic-quintic nonlinearity

Results of a comprehensive dynamical analysis are reported for several fundamental species of bright solitons in the one-dimensional lattice modeled by the discrete nonlinear Schrödinger equation with the cubic-quintic nonlinearity. Staggered solitons, which were not previously considered in this model, are studied numerically, through the computation of the eigenvalue spectrum for modes of small perturbations, and analytically, by means of the variational approximation. The numerical results confirm the analytical predictions. The mobility of discrete solitons is studied by means of direct simulations, and semianalytically, in the framework of the Peierls-Nabarro barrier, which is introduced in terms of two different concepts, free energy and mapping analysis. It is found that persistently moving localized modes may only be of the unstaggered type.

Cubic-quintic solitons in the checkerboard potential

We introduce a two-dimensional (2D) model which combines a checkerboard potential, alias the Kronig-Penney (KP) lattice, with the self-focusing cubic and self-defocusing quintic nonlinear terms. The beam-splitting mechanism and soliton multistability are explored in this setting, following the recently considered 1D version of the model. Families of single- and multi-peak solitons (in particular, five- and nine-peak species naturally emerge in the 2D setting) are found in the semi-infinite gap, with both branches of bistable families being robust against perturbations. For single-peak solitons, the variational approximation (VA) is developed, providing for a qualitatively correct description of the transition from monostability to the bistability. 2D solitons found in finite band gaps are unstable. Also constructed are two different species of stable vortex solitons, arranged as four-peak patterns ("oblique" and "straight" ones). Unlike them, compact "crater-shaped" vortices are unstable, transforming themselves into randomly walking fundamental beams.