Intrinsic localized modes in coupled DNLS equations from the anti-continuum limit

Abstract

In the present work, we generalize earlier considerations for intrinsic localized modes consisting of a few excited sites, as developed in the one-component discrete nonlinear Schrödinger equation model, to the case of two-component systems. We consider all the different combinations of "up" (zero phase) and "down" (π phase) site excitations and are able to compute not only the corresponding existence curves, but also the eigenvalue dependence of the small eigenvalues potentially responsible for instabilities, as a function of the nonlinear parameters of the model representing the self/cross phase modulation in optics and the scattering length ratios in the case of matter waves in optical lattices. We corroborate these analytical predictions by means of direct numerical computations. We infer that all the modes which bear two adjacent nodes with the same phase are unstable in the two component case and the only solutions that may be linear stable are ones where each set of adjacent nodes, in each component is out of phase.

Discrete solitons dynamics in $$\mathscr {PT}$$-symmetric oligomers with complex-valued couplings

“…This proposed model is integrable and its integrability is further utilized to build up the phase portrait of the system. The existence and stability of localized mode solutions to nonlinear dynamical lattices of the DNLS type of equations with two-component settings have been considered and a general framework has been provided in [64]. A dual-core nonlinear waveguide with the PT -symmetry has been expanded by including a periodic sinusoidal variation of the loss-gain coefficients and synchronous variation of the inter-core coupling constant [65].…”