Abstract
We consider a two-component one-dimensional model of gap solitons (GSs), which is based on two nonlinear Schrödinger equations, coupled by repulsive XPM (cross-phase-modulation) terms, in the absence of the SPM (self-phase-modulation) nonlinearity. The equations include a periodic potential acting on both components, thus giving rise to GSs of the "symbiotic" type, which exist solely due to the repulsive interaction between the two components. The model may be implemented for "holographic solitons" in optics, and in binary bosonic or fermionic gases trapped in the optical lattice. Fundamental symbiotic GSs are constructed, and their stability is investigated, in the first two finite bandgaps of the underlying spectrum. Symmetric solitons are destabilized, including their entire family in the second bandgap, by symmetry-breaking perturbations above a critical value of the total power. Asymmetric solitons of intra-gap and inter-gap types are studied too, with the propagation constants of the two components falling into the same or different bandgaps, respectively. The increase of the asymmetry between the components leads to shrinkage of the stability areas of the GSs. Inter-gap GSs are stable only in a strongly asymmetric form, in which the first-bandgap component is a dominating one. Intra-gap solitons are unstable in the second bandgap. Unstable two-component GSs are transformed into persistent breathers. In addition to systematic numerical considerations, analytical results are obtained by means of an extended ("tailed") Thomas-Fermi approximation (TFA).
Highlights
Studies of solitons in spatially periodic potentials have grown into a vast area of research, with profoundly important applications to nonlinear optics, plasmonics, and matter waves in quantum gases, as outlined in recent reviews [1,2,3,4]
Both the experimental and theoretical results reveal that solitons can be created in lattice potentials, if they do not exist in the uniform space [this is the case of gap solitons (GSs) supported by the self-defocusing nonlinearity, see original works [13,14,15,16] and reviews [7, 17]], and solitons may be stabilized, if they are unstable without the lattice
Almost the entire symmetric family is stable against symmetric perturbations, i.e., it is stable in the framework of the single-component equation, except for a weak oscillatory instability, accounted for by quartets of complex-conjugate eigenvalues, in the form of λ = ±iIm (λ ) ± Re (λ ), which appears near the left edge of the second bandgap— namely, at k < kmin ≈ −3.45
Summary
Studies of solitons in spatially periodic (lattice) potentials have grown into a vast area of research, with profoundly important applications to nonlinear optics, plasmonics, and matter waves in quantum gases, as outlined in recent reviews [1,2,3,4]. Parallel to the progress in the experiments, the study of the interplay between the nonlinearity and periodic potentials has been an incentive for the rapid developments of theoretical methods [11, 12] Both the experimental and theoretical results reveal that solitons can be created in lattice potentials, if they do not exist in the uniform space [this is the case of gap solitons (GSs) supported by the self-defocusing nonlinearity, see original works [13,14,15,16] and reviews [7, 17]], and solitons may be stabilized, if they are unstable without the lattice The stability of GSs has been studied in detail too—close to edges of the corresponding bandgaps—in one [27,28,29] and two [30] dimensions alike
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