Spatial properties of integrable and nonintegrable discrete nonlinear Schrödinger equations

Spatial structure of the non-integrable discrete defocusing Hirota equation

In this paper, we investigate the spatial property of the non-integrable discrete defocusing Hirota equation utilizing a planar nonlinear discrete dynamical map method. We construct the periodic orbit solutions of the stationary discrete defocusing Hirota equation. The behavior of the orbits in the vicinity of the special periodic solution is analyzed by taking advantage of the named residue. We characterize the effects of the parameters on the aperiodic orbits with the aid of numerical simulations. A comparison with the non-integrable discrete defocusing nonlinear Schrödinger equation case reveals that the non-integrable discrete defocusing Hirota equation has more abundant spatial properties. Rather an interesting and novel thing is that for any initial value, there exists triperiodic solutions for a reduced map.

Nonintegrable spatial discrete nonlocal nonlinear schrödinger equation

Integrable and nonintegrable discrete nonlinear Schrödinger equations (NLS) are significant models to describe many phenomena in physics. Recently, Ablowitz and Musslimani introduced a class of reverse space, reverse time, and reverse space-time nonlocal integrable equations, including the nonlocal NLS equation, nonlocal sine-Gordon equation, nonlocal Davey-Stewartson equation, etc. Moreover, the integrable nonlocal discrete NLS has been exactly solved by inverse scattering transform. In this paper, we study a nonintegrable discrete nonlocal NLS, which is a direct discrete version of the reverse space nonlocal NLS. By applying discrete Fourier transform and modified Neumann iteration, we present its stationary solutions numerically. The linear stability of the stationary solutions is examined. Finally, we study the Cauchy problem for the nonlocal NLS equation numerically and find some different and new properties on the numerical solutions comparing with the numerical solutions of the Cauchy problem for the NLS equation.

Nonintegrable semidiscrete Hirota equation: gauge-equivalent structures and dynamical properties

In this paper, we investigate nonintegrable semidiscrete Hirota equations, including the nonintegrable semidiscrete Hirota(-) equation and the nonintegrable semidiscrete Hirota(+) equation. We focus on the topics on gauge-equivalent structures and dynamical behaviors for the two nonintegrable semidiscrete equations. By using the concept of the prescribed discrete curvature, we show that, under the discrete gauge transformations, the nonintegrable semidiscrete Hirota(-) equation and the nonintegrable semidiscrete Hirota(+) equation are, respectively, gauge equivalent to the nonintegrable generalized semidiscrete modified Heisenberg ferromagnet equation and the nonintegrable generalized semidiscrete Heisenberg ferromagnet equation. We prove that the two discrete gauge transformations are reversible. We study the dynamical properties for the two nonintegrable semidiscrete Hirota equations. The exact spatial period solutions of the two nonintegrable semidiscrete Hirota equations are obtained through the constructions of period orbits of the stationary discrete Hirota equations. We discuss the topic regarding whether the spatial period property of the solution to the nonintegrable semidiscrete Hirota equation is preserved to that of the corresponding gauge-equivalent nonintegrable semidiscrete equations under the action of discrete gauge transformation. By using the gauge equivalent, we obtain the exact solutions to the nonintegrable generalized semidiscrete modified Heisenberg ferromagnet equation and the nonintegrable generalized semidiscrete Heisenberg ferromagnet equation. We also give the numerical simulations for the stationary discrete Hirota equations. We find that their dynamics are much richer than the ones of stationary discrete nonlinear Schrödinger equations.

Elastic scattering of traveling phonons by discrete breathers as a time-periodic scatterer in the one-dimensional discrete nonlinear Schrödinger model: Exact analysis

Discrete nonlinear Schrödinger model is a nonlinear lattice model used to investigate different nonlinear phenomena arising in many physical contexts. In this work we used this model to observe linear traveling phonon scattering by time-periodic discrete breathers in the piecewise smooth (PWS) version of nonlinearity. In one-dimensional system the single-channel scattering is found to be elastic (with the incoming and outgoing fluxes of energy being equal to each other). Considering one-site symmetric breather solution we are able to calculate the exact expression for the transmission coefficient (T) or the ratio of the transmitted to incident flux of energy in such a system using transfer-matrix formalism. From the condition of perfect transmission ( T=1 , and hence reflection coefficient R=0 ) or the ratio of reflected to incident flux of energy for elastic scattering is zero, our observation shows that perfect transmission happens to appear at the threshold of a localized mode, which occurs at the band edge of the extended eigenmodes (plane waves). We have also presented the results obtained from the condition of perfect reflection. The advantage of using the PWS version of nonlinearity in the model is that all the results derived are exact. Numerical simulations complement our results.

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.