Vortex and dipole solitons in complex two-dimensional nonlinear lattices

Abstract

Using computational methods, it is found that the two-dimensional nonlinear Schrödinger (NLS) equation with a quasicrystal lattice potential admits multiple dipole and vortex solitons. The linear and the nonlinear stability of these solitons is investigated using direct simulations of the NLS equation and its linearized equation. It is shown that certain multiple vortex structures on quasicrystal lattices can be linearly unstable but nonlinearly stable. These results have application to investigations of localized structures in nonlinear optics and Bose-Einstein condensates.

Vortex Structures in Saturable Media

“…Previously in Ref. [23], it is numerically demonstrated that, for the cubic nonlinearity case, the maximum amplitude of the four-hump vortex solitons centered at the periodic lattice maxima increases sharply, after z ¼ 0:3 and due to this instant blow up in the maximum amplitude these vortex solitons are found to be nonlinearly unstable. It is shown that the phase structure also breaks up for the same vortex structure.…”

“…We should note that, for cubic nonlinearity case, in Ref. [23], the nonlinear stability properties of the dipole solitons on both periodic and the Penrose lattice maxima are demonstrated and both dipole solitons are found to be nonlinearly unstable since they exhibit strong localization after a few diffraction lengths and breakup in their phase structures. The Penrose lattice dipole solitons also suffer from the drift instability since the dipole humps both move from the lattice maxima toward nearby lattice minima immediately.…”

“…Since the lattice solitons located on lattice maxima are found to be nonlinearly unstable for cubic nonlinearity case in Ref. [23], in this work, we deal with the dipoles and vortex solitons located on lattice maxima to demonstrate the stabilization effect of the saturation.…”

“…In Kerr media, in Ref. [23], it is computationally demonstrated that the twodimensional cubic nonlinear Schrödinger (NLS) equation with a quasicrystal lattice potential admits multiple dipole and vortex solitons. In aforementioned work, it is numerically shown that certain multiple vortex structures on quasicrystal lattices tend to be nonlinearly stable if the vortex/dipole humps are located on lattice minima but they are nonlinearly unstable if the humps are located on lattice maxima.…”

Time-dependent spectral renormalization method

“…(38) can lead to exponential growth or decay which will eventually put serious strain on the time numerical integration. To incorporate physics into (38) we replace |R(τ )| 2 that appears inside the exponent by any of the expressions given in Table 1 where…”