Abstract
We modified the spectral renormalization method as the pseudospectral renormalization method in order to find the localized solutions. The pseudospectral renormalization method can be applied to a large class of problems including different homogeneities. Using this computational method, we demonstrate the existence of two different solitons in optical media described by the self-focusing cubic and the self-defocusing quintic nonlinear Schrödinger equation with quasicrystal lattice. It is shown that there are two different lattice solitons corresponding to the first and the second renormalization factors for the self-focusing cubic and the self-defocusing quintic model. However, the self-focusing quintic nonlinearity without optical lattice does not support two different solitons. We showed that the lattice solitons corresponding to the first and the second renormalization factors have the same powers and amplitudes. We also demonstrate that quintic nonlinearity supports bistable solitons by adding the optical lattice such as a quasicrystal lattice. The linear and nonlinear stabilities of these solitons are investigated using direct simulation of the nonlinear Schrödinger equation with the cubic-quintic nonlinearity and its linearized equation.
Highlights
Optical solitons, that is, localized waves, maintain their profile in nonlocal optical media due to the balance between the group-velocity dispersion, diffraction, and nonlinear selfmodulation
We demonstrate the existence of two different solitons in optical media described by the selffocusing cubic and the self-defocusing quintic nonlinear Schrodinger equation with quasicrystal lattice
We demonstrate that quintic nonlinearity supports bistable solitons by adding the optical lattice such as a quasicrystal lattice
Summary
That is, localized waves, maintain their profile in nonlocal optical media due to the balance between the group-velocity dispersion, diffraction, and nonlinear selfmodulation. Atomic crystals can have various irregularities, such as defects and edge dislocations, and quasicrystal structures, which have long-range orientational order but no translational symmetry [12, 13] The optical lattices, such as periodic and quasicrystal lattices, are not necessary for stability of the solitons in the self-focusing cubic media [14]. The nonlinear Schrodinger equation with the cubic and quintic nonlinearity is universal mathematical model describing many physical situations. The nonlinear and linear stability properties of these solitons are studied by direct computations of nonlinear Schrodinger equation with cubic-quintic nonlinearities where the initial conditions are taken to be the lattice solitons with %1 perturbation
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