Multi-soliton states under triangular spatial modulation of the quadratic nonlinearity

Abstract

We introduce multi-soliton sets in the two-dimensional medium with the χ (2) nonlinearity subject to spatial modulation in the form of a triangle of singular peaks. Various families of symmetric and asymmetric sets are constructed, and their stability is investigated. Stable symmetric patterns may be built of 1, 4, or 7 individual solitons, while stable asymmetric ones contain 1, 2, or 3 solitons. Symmetric and asymmetric patterns may demonstrate mutual bistability. The shift of the asymmetric single-soliton state from the central position is accurately predicted analytically. Vortex rings composed of three solitons are produced too. possibility to generalize the study of the onset of collapse in nonlinear media [12,13], as well as to emulate the nonlinear dynamics in a sub-1D space, with the effective dimension D = 2 (1 − α) / (2 − α) < 1 [11]. The singular modulation can be emulated by means of above-mentioned techniques, tuning them to the exact resonance in a narrow layer.Another possibility is to consider spatial modulation of the local interaction strength in media with the quadratic (χ (2) ) nonlinearity, which has well-known realizations in optics [14]- [17]. Experimental realizations of such settings are possible, in particular, using the well-elaborated technique of the quasi-phasematching [18]- [20], which can be implemented in a spatially nonuniform form, in 1D and 2D geometries alike, thus helping one to create a required profile of the χ (2) coefficient [21]-[23]. In the theoretical analysis, singular modulation of the quadratic nonlinearity in the 1D system, accounted for by a delta-function,, and localized modes (solitons) pinned to it, were introduced in Ref. [24], and a pair of modulating delta-functions was considered in Ref. [25]. A discrete version of the localized quadratic nonlinearity was elaborated in the form of a linear lattice with one or two χ (2) -nonlinear sites embedded in it [26].While the delta-like modulation of the local χ (2) coefficient may not be easily realized in the experiment, a more realistic case of the 1D singular modulation, with χ (2) ∼ |x| −α and positive α, was introduced in Ref. [27]. It was found that this modulation format supports quadratic solitons, pinned to the singular peak, for α < 1 (the pinned modes vanish at α = 1), and they are chiefly stable. A natural extension of that setting is a symmetric pair of two peaks (similar to the above-mentioned symmetric set of two delta-functions multiplying the nonlinear terms [25]). The consideration of the twin peaks has made it possible to predict effects of the spontaneously symmetry breaking [28] and formation of asymmetric two-soliton states pinned to the two peaks [27]. These results may be used for the design of steering optical beams in the form of spatial solitons.An essential advantage of using the quadratic nonlinearity with this type of the local modulation is that it may be extended to the 2D geometry, by choosing χ (2) (r) ∼ r −α (r is the radial coordinate), while any 2D singular modulation of the self...

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