Flat-top (FT) solitons are optical pulses that arise from the balance of dispersion and self-phase modulation in media with the competing cubic-quintic nonlinearity. Previously, FT solitons were studied only in the case of the second-order dispersion ($m=2$). Following the recent observation of pure-quartic solitons (corresponding to $m=4$), we here construct families of FT solitons in the setting with pure-high-even-order dispersion (PHEOD), including $m=4,6,8$, and $10$, and address interactions between them. The PHEOD solitons are completely stable, and, unlike the conventional solitons, they feature oscillatory tails. Interactions between the PHEOD solitons are anomalous, featuring repulsion and attraction between in- and out-of-phase solitons, respectively. These results expand the variety of optical solitons maintained by diverse dispersive nonlinear media.
Much interest to flat-top (FT) solitons was drawn by the experimental realization of quantum droplets [1,2] in dipolar [3][4][5] and binary [6][7][8][9] Bose-Einstein condensates (BECs) in ultracold atomic gases. FT solitons are produced by nonlinear Schrödinger equations (NLSEs) with competing nonlinearities. In BEC mixtures, competing nonlinearities originate from the interplay of the meanfield inter-atomic interactions and Lee-Huang-Yang correction induced by quantum fluctuations [10,11], which leads to the creation of FT modes in the form of quantum droplets [12,13]. In nonlinear optics, competing nonlinearities usually combine the self-focusing cubic and defocusing quintic terms, which provides the stabilization of two-and three-dimensional solitons against the collapse driven by the self-focusing [14][15][16][17][18][19][20][21] .
Recently, a new class of soliton pulses, known as purequartic solitons (PQSs), has been identified, arising from the balance between the fourth-order group-velocity dispersion (GVD) and self-phase modulation (SPM). PQSs have been predicted and experimentally observed in photonic-crystal waveguides by means of precise GVD engineering [22], using a mode-locked laser incorporating an intra-cavity spectral pulse shaper [23,24]. The conventional solitons, supported by the second-order GVD, and PQSs are just the two lowest-order members of an infinite hierarchy of soliton pulses arising from the interplay of the Kerr nonlinearity and negative pure-high-evenorder (PHEOD) dispersion. Experimental studies indicate that more general PHEOD solitons arise from the * lpf281888@gmail.com balance between the SPM and a single negative PHEOD dispersion term of orders m = 6, 8, and 10 [25]. In subsequent research, PHEOD solitons have been demonstrated to form bound states [26,27].
Unlike the conventional solitons, PQSs and PHEOD solitons feature oscillations in their exponentially decaying tails [28]. Importantly, the PQSs obey the scaling relation with the energy proportional to the third power of the inverse pulse duration, compared to the conventional solitons, whose energy scales as the inverse of the pulse duration, τ -1 0 . Generally, the energy of PHEOD solitons scales as τ -(m-1) 0 , implying that they obey a favorable energy scaling, making it possible to attain higher energies than the conventional GVD solitons. Another difference is that the PQSs and PHEOD solitons do not obey the Galilean invariance, making the creation of moving solitons a nontrivial issue [29].
Recent advances in the work with PQSs and PHEOD solitons suggest a possibility for the development of a new branch of nonlinear optics and may lead to novel applications [30,31]. Recently, several species of such solitons have been numerically investigated in the framework of the NLSE, including dissipative [32,33], Raman [34][35][36], dark [37,38], and spatiotemporal pure-quartic [40] solitons, as well as pure-quartic domain walls [39] and bound states of PQSs (“molecules”) [41,42]. However, PQSs and PHEOD solitons of the FT type have not yet been investigated. In this work we focus on FT solitons produced by the NLSE with the PHEOD terms and competing cubic-quintic (CQ) nonlinearity. Our objective is to reveal properties of the PHEOD solitons of the FT type and their interactions. By means of numerically methods, we produce families of FT solitons supported by GVD of orders m = 4, 6, 8, and 10. Our study shows anomalous interactions of such solitons. These findings help to further understand the physical purport of the PHEOD solitons and extend the variety of optical solitons in dispersive nonlinear media.
The starting point is the modified one-dimensional NLSE which governs the propagation of optical pulses in the medium with PHEOD and the CQ nonlinearity:
where Ψ (z, τ ) is the field envelope, z is the propagation distance, and τ is the retarded time in the reference frame moving with the group velocity of the carrier wave. We consider the anomalous sign of PHEOD of even orders m = 4, 6, 8, 10.
We look for stationary PHEOD solutions to Eq. ( 1) with a real temporal profiles ψ and propagation constant β as Ψ (z, τ ) = e iβz ψ(τ ).
(
The substitution of this in Eq. ( 1) leads to the equation for ψ:
We solved Eq. ( 3) numerically, using the Newtonconjugate gradient method [43]. With an appropriate initial guess, the method converges to numerical solutions through successive iterations.
Families of PHEOD solitons are characterized by their energy (alias the integral norm),
which is a dynamical invariant of Eq. ( 1). The numerically obtained dependence of their propagation constant β on energy E is plotted, on the logarithmic scale, in Fig. 1(a). To compare FT solitons produced by Eq. ( 3) with the conventional FT solitons, known from the solution of the second-order NLSE with the CQ nonlinearity, the curve β(E) for m = 2 is included too, showing that it is essentially different from its counterparts for m ≥ 4.
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