Gradient catastrophe of nonlinear photonic valley-Hall edge pulses
Daria A. Smirnova, Lev A. Smirnov, Ekaterina O. Smolina, Dimitris G. Angelakis, Daniel Leykam
GGradient catastrophe of nonlinear photonic valley-Hall edge pulses
Daria A. Smirnova,
1, 2
Lev A. Smirnov,
2, 3
Ekaterina O. Smolina, Dimitris G. Angelakis,
4, 5 and Daniel Leykam Nonlinear Physics Centre, Australian National University, Canberra ACT 2601, Australia Institute of Applied Physics, Russian Academy of Science, Nizhny Novgorod 603950, Russia Nizhny Novgorod State University, Gagarin Av. 23, Nizhny Novgorod, 603950 Russia Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 School of Electrical and Computer Engineering, Technical University of Crete, Chania, Greece 73100
We derive nonlinear wave equations describing the propagation of slowly-varying wavepacketsformed by topological valley-Hall edge states. We show that edge pulses break up even in theabsence of spatial dispersion due to nonlinear self-steepening. Self-steepening leads to the previously-unattended effect of a gradient catastrophe, which develops in a finite time determined by the ratiobetween the pulse’s nonlinear frequency shift and the size of the topological band gap. Taking theweak spatial dispersion into account results then in the formation of stable edge quasi-solitons. Ourfindings are generic to systems governed by Dirac-like Hamiltonians and validated by numericalmodeling of pulse propagation along a valley-Hall domain wall in staggered honeycomb waveguidelattices with Kerr nonlinearity.
The combination of topological band structureswith mean field interactions not only gives rise to richnonlinear wave physics [1, 2], but is also anticipatedto unlock advanced functionalities, such as magnet-free nonreciprocity [3], tunable and robust waveguid-ing [4, 5], and novel sources of classical and quan-tum light [6–11]. Valley-Hall photonic crystals [12–24]show great promise for these applications, due to theirability to combine slow-light enhancement of nonlin-ear effects with topological protection against disor-der, which limits the performance of conventional pho-tonic crystal waveguides [25–27].Accurate modelling of pulse propagation throughphotonic crystal waveguides in the slow light regimerequires taking into account the dispersion in the ef-fective nonlinearity strength, which can induce effectssuch as pulse self-steepening and supercontinuum gen-eration [28–31]. Despite the proven importance ofthese effects in applications [32–34], analysis of non-linear light propagation in topological photonic struc-tures most often assumes non-dispersive nonlineari-ties, in both the underlying material response [35–45] and effective models describing the propagationof nonlinear edge states [46–50]. The simplest effec-tive model is the cubic nonlinear Schr¨odinger equation(NLSE), which describes the self-focusing dynamics ofedge wavepackets independent of the properties of thetopological band gap, such as its size.More sophisticated effective models such as nonlin-ear Dirac models explicitly include the nontrivial spin-like degrees of freedom required to create topologicalband gaps [51–55]. In the bulk, the nonlinear Diracmodel supports self-induced domain walls and solitonswhose stability and degree of localization are sensitiveto the gap size. Infinitely-extended (plane wave-like)nonlinear edge states can be obtained analytically andexhibit similar features. However, the nonlinear dy-namics of localized edge pulses within the nonlinearDirac model framework were not yet considered.In this paper we study an analytically-solvablenonlinear Dirac model (NDM) describing topological edge pulses, revealing that nonlinear topological edgestates’ exhibit a self-steepening nonlinearity when thepulse self-frequency modulation becomes comparableto the width of the topological band gap. The self-steepening nonlinearity leads to the formation of agradient catastrophe of edge wavepackets within a fi-nite propagation time proportional to the pulse width.Taking the weak spatial dispersion of the topologicaledge modes into account regularizes the catastropheand results in the formation of stable edge solitonsfor sufficiently long pulses. We validate our analysisusing numerical simulations of beam propagation ina laser-written valley-Hall waveguide lattice, demon-strating that this effect should be observable even forrelatively weak nonlinearities. Our findings suggestvalley-Hall photonic crystal waveguides will providea fertile setting for observing and harnessing higher-order nonlinear optical effects.We consider a generic continuum Dirac model oftopological photonic lattices with incorporated non-linear terms. The evolution of a spinor wavefunction Ψ ( x, y, t ) = [Ψ ( x, y, t ); Ψ ( x, y, t )] T in the vicinity ofa band crossing point (topological phase transition) isgoverned by the nonlinear Dirac equation [51, 54, 55] i∂ t Ψ = (cid:16) ˆ H D ( δ k ) + ˆ H NL (cid:17) Ψ ; (1)ˆ H D ( δ k ) = δk x ˆ σ x + δk y ˆ σ y + M ˆ σ z , (2)where δ k = ( δk x , δk y ) ≡ − i ( ∂ x , ∂ y ) is the momen-tum, M is a detuning between two sublattices or spinstates, and ˆ H NL = − g diag[ | Ψ | ; | Ψ | ] is a local non-dispersive Kerr nonlinearity.As a specific example, the model (1) can be imple-mented in nonlinear photonic graphene with a stag-gered sublattice potential as illustrated in Fig. 1(a).A dimerized graphene lattice is composed of single-mode dielectric waveguides with local Kerr nonlinear-ity. The effective mass M characterises a detuning be-tween propagation constants in the waveguides of twosublattices. The form of Hamiltonian operator (2)assumes normalization of the transverse coordinates a r X i v : . [ phy s i c s . op ti c s ] F e b x, y and evolution variable in the propagation direc-tion t ∼ z/v D to the Dirac velocity v D = 3 κa / κ and a distance a between two neighboring waveg-uides [23, 55]. This continuum model is valid provided | κ | (cid:38) | M | [23].The valley-Hall domain wall is created between twoinsulators characterized by different signs of the mass.We take M ( y >
0) = M and M ( y <
0) = − M and without loss of generality assume M > y = 0: Ψ ( y, x, t ) = [ ψ ( y, ω, k ); ψ ( y, ω, k )] T e ikx − iωt . Basedon the close connections between nonlinear edge statesand self-trapped Dirac solitons in topological bandgaps revealed in Ref. [55], the nonlinear edge modedispersion can be obtained as ω + k = − g I / I = | ψ , ( y =0) | (see Supplemental Material [56]). In Figs. 1(b,c)we show the plane wave-like profile of the nonlinearedge mode with fixed wavenumber k parallel to theedge.Using the global parity symmetry with respect tothe interface and analytical solutions for the edgestates [56], we calculate two characteristics of the edgestates via integration in the upper half plane y > FIG. 1. (a) Dimerized photonic graphene lattice withstaggered sublattice potential. Here κ denotes the tun-neling coefficient between waveguides, a is the distancebetween neighboring waveguides, and the round arrow g illustrates the nonlinear self-action effect. (b) Trans-verse profile I ( y ) and (c) in-plane intensity distribution I ( x, y ) = | ψ ( x, y ) | + | ψ ( x, y ) | of the nonlinear edgewave propagating along the x axis and bound to the do-main wall located at y = 0, where the effective mass M ( y )changes sign. Parameters are M = 1 , g = 0 . the power P and spin S x , P = (cid:90) ∞ ψ † ψ dy, S x = 12 (cid:90) ∞ ψ † ˆ σ x ψ dy , (4)and identify a functional relation S x ( P ) betweenthem: S x = − g arcsin (cid:20) √ (cid:18) P g √ (cid:19)(cid:21) . (5)Crucially, this relation is independent of the wavevec-tor k , which allows us to develop a slowly varyingenvelope approximation to describe the nonlinear dy-namics of finite edge wavepackets. Using Eq. (1), itcan be shown that the integral characteristics obeythe following evolution equation: ∂ t P = − ∂ x S x ( P ) . (6)Next, we assume P ( x, t ) and S x ( x, t ) are slowly vary-ing functions of the local frequency and wavenumber,such that Eq. (5) remains valid to a first approxima-tion for smooth x -dependent field envelopes. PluggingEq. (5) into Eq. (6), and using Eq. (3) assuming weaknonlinearity g I (cid:28) M , we obtain the simple nonlin-ear wave equation for the longitudinal intensity profile I ( x, t ): ∂ t I − ∂ x I (cid:0) − g I / (cid:0) M (cid:1)(cid:1) = 0 . (7)Equation (7) suggests that the evolution of finitewavepackets propagating along the x axis is accom-panied by steepening of the trailing wavefront up tothe development of a gradient catastrophe . Note, inthe linear case g = 0, Eq. (7) shows that the edgewavepacket of any shape does not diffract and prop-agates along the domain wall with constant groupvelocity v = −
1, being granted with topological ro-bustness. Alternatively, Eq. (7) can be derived usingasymptotic methods based on a series expansion ofthe spinor wavefunction [56]Ψ , ( x, y, t ) = ± a ( ξ ; τ n ) e − M | y | + ∞ (cid:88) n =1 µ n Ψ ( n )1 , ( y ; ξ ; τ n ) , (8)where we have introduced a small parameter µ ∼ g I /M , a hierarchy of time scales: τ n = µ n t , and as-sumed a smooth dependence of the spinor componentson t in the moving coordinate frame ( ξ ≡ x + t, y ).To illustrate the key effect of the gradient catas-trophe captured by Eq. (7), we model the timedynamics of an edge wavepacket using a cus-tom numerical code, applying a split-step schemeand the fast Fourier transform to solve Eq. (1).Figure 2 shows evolution of the initial distribu-tion set in the form of the edge state across theinterface with the Gaussian envelope I ( x, t =0) = F e − x / Λ along the x axis: ψ ( y, x, t = 0) = (cid:2) ψ (cid:0) y, ω = − g I ( x ) / , (cid:1) ; ψ (cid:0) y, ω = − g I ( x ) / , (cid:1)(cid:3) T .Plugging the Gaussian distribution into Eq. (7), wemay estimate the pulse breakdown time t ∗ analyti-cally: t ∗ = 2 √ e Λ (cid:18) M gF (cid:19) . (9)Thus, pulse breakdown occurs for finite wavepacketswhen the peak nonlinear frequency shift becomes com-parable to the size of the topological band gap. As thepulse propagates its tail becomes increasingly steep,developing a discontinuity (i.e. a shock) in a finitetime. The numerical solution of Eq. (1) is fully con-sistent with our analytical considerations, see Fig. 2.Weak spatial dispersion effects serve as a possiblemechanism regularizing the gradient catastrophe, re-sulting in the formation of solitons. For honeycombphotonic lattices, dispersion is accounted for by intro-ducing off-diagonal second-order derivatives with thecoefficient η = (6 κ ) − into the Dirac model (1):ˆ H disp = (cid:18) − η ( − i∂ x + ∂ y ) − η ( i∂ x + ∂ y ) (cid:19) . (10)Assuming ηM ∼ µ and developing a perturba-tion theory with expansion (8), we derive an evolu-tion equation governing the complex-valued amplitude a ( ξ, t ): i (cid:18) ∂ t a + g | a | M a∂ ξ | a | (cid:19) + g | a | a + η (cid:0) ∂ ξ a − M a (cid:1) = 0 , (11) FIG. 2. Gradient catastrophe development. The Gaussianpulse with F = 1, Λ = 3 . t = 0. Slicescolor code intensity distributions I ( x, y ) at the given mo-ments: (a) t = 20, (b) t = 40, (c) t = 60. Cuts alongthe domain wall at y = 0 show consistency of the numeri-cal solution (blue curves) with the solution of the nonlin-ear simple wave equation (7) for the intensity (red dottedlines). Parameters are M = 1, g = 0 .
75. In Figs. 2, 3,dashed lines trace the domain wall separating spatial do-mains with masses of the opposite sign as indicated byshading with different colors on the top surface. which differs from the conventional cubic nonlinearSchr¨odinger equation by the second higher-order non-linear term responsible for the phase modulation andself-steepening. This equation enables analysis of boththe modulational instability of nonlinear plane-wave-like edge states, and the formation of edge quasi-solitons [56].To verify the validity of the modified NLSE (11)we consider the propagation of a Gaussian pulse inFig. 3. The conventional NLSE, which lacks the self-steepening term, only exhibits self-focusing and grad-ual self-compression of the pulse. On the other hand,the modified Eq. (11) correctly reproduces the grow-ing asymmetry of the edge pulse as it propagates.We show in the Supplemental Material how the self-steepening leads to the break-up of wide pulses, re-sulting in the radiation of part of its energy into bulkmodes, with the remainder forming a quasi-solitonwhich continues to propagate along the edge and iscapable of traversing sharp bends. We note that evenafter the initial pulse breakup, self-steepening termscan influence the soliton stability and soliton-solitoninteractions [32].As a possible implementation, we consider a realis-tic valley-Hall waveguide array of laser-written waveg-uides with parameters similar to those used in theexperimental work Ref. [16]. In this case, the evolu-tion variable t becomes the longitudinal propagationdistance z . To simulate the evolution, we solve theparaxial equation numerically in a periodic potentialby propagating an optical wavepacket [56]. For realis-tic laser input powers we observe a notable distortionof the beam and signatures of the catastrophe devel-opment at its trailing wavefront [Fig. 4(a,b)]. Fig-ure 4(c) plots the intensity map in xz interface plane.The rapidly developing asymmetry at short distances FIG. 3. Nonlinear pulse transformation in the domain-wall problem with dispersion. Snapshots show intensitydistributions in-plane I ( x, y ) (top row) and along the do-main wall I ( x, y = 0) (bottom row) at the given times:(a) t = 22, (b) t = 66, (c) t = 110. Overlaid curvesare the pulse envelopes calculated by using NLSE (greendashed) and Eq. (11) (red dotted). The Gaussian pulsewith F = 0 .
18, Λ = 5 / √ t = 0. Parame-ters are M = 1, g = 1, η = 0 . FIG. 4. Nonlinear dynamics of the optical beam at thevalley-Hall domain wall at a zigzag interface in a honey-comb lattice of laser-written waveguides. The input beamhas Gaussian envelope along the domain wall with max-imum intensity I = 2 . × W/m . (a,b) Intensitydistributions of the nonlinear beam (top panels, blue line)at propagation distances (a) z = 11 mm (left column) and(b) z = 22 mm (right column). For comparison, the lin-ear beam evolution, i.e. with nonlinearity switched off, isshown on bottom panels and with dotted red lines. Pur-ple arrows points to the direction of motion. Dashed greyline depicts the domain wall. (c) Intensity distribution atthe interface obtained in modeling of the paraxial equation(left) and the corresponding Dirac model (right). The pur-ple dashed line traces a straight trajectory of the center ofmass of the linear pulse. (d) Breakdown coordinate z ∗ as afunction of the input intensity I estimated from the Diracmodel (green curve) and paraxial modeling (cyan crosses).The dashed grey lines’ intersection indicates the value ofthe input beam intensity used for (a,b,c). agrees with modeling of the corresponding continuumDirac equations and estimates of the breakdown coor-dinate z ∗ based on Eq. (9) [Fig. 4(d)].In conclusion, we have described the gradient catas-trophe of the nonlinear edge wavepackets in thespinor-type Dirac equation and the formation ofedge solitons at the valley-Hall domain walls. Wehave derived a higher-order self-steepening nonlinearSchr¨odinger equation describing these effects. Spa-tiotemporal numerical modeling confirmed that pulseself-steepening can manifest already in the frameworkof paraxial optics in weakly nonlinear media, such astopological waveguide lattices, and will likely play akey role in future experiments with topological pho-tonic crystal waveguides. Beyond the specific valley-Hall example we considered, our findings are instruc-tive for other emerging experimental studies of nonlin-ear dynamic phenomena in topological systems, suchas the Chern insulators and their implementations in a variety of physical platforms spanning from meta-materials [4] to optical lattices [43, 44] and exciton-polariton condensates [57].This work was supported by the Australian Re-search Council (Grant DE190100430), the RussianFoundation for Basic Research (Grant 19-52-12053),the National Research Foundation, Prime MinistersOffice, Singapore, the Ministry of Education, Singa-pore under the Research Centres of Excellence pro-gramme, and the Polisimulator project co-financedby Greece and the EU Regional Development Fund.Theoretical analysis of the continuum model was sup-ported by the Russian Science Foundation (Grant No.20-72-00148). D.A.S. thanks Yuri Kivshar for valuablediscussions. [1] D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar,“Nonlinear topological photonics,” Applied PhysicsReviews , 021306 (2020).[2] A. Saxena, P. G. Kevrekidis, and J. Cuevas-Maraver, Nonlinearity and Topology (Springer InternationalPublishing, Cham, 2020), pp. 25–54.[3] W. Chen, D. Leykam, Y. Chong, and L. Yang, “Non-reciprocity in synthetic photonic materials with non-linearity,” MRS Bulletin , 443–451 (2018).[4] D. A. Dobrykh, A. V. Yulin, A. P. Slobozhanyuk,A. N. Poddubny, and Y. S. Kivshar, “Nonlinear con-trol of electromagnetic topological edge states,” Phys.Rev. Lett. , 163901 (2018).[5] M. I. Shalaev, W. Walasik, and N. M. Litchinitser,“Optically tunable topological photonic crystal,” Op-tica , 839 (2019).[6] S. Kruk, A. Poddubny, D. Smirnova, L. Wang,A. Slobozhanyuk, A. Shorokhov, I. Kravchenko,B. Luther-Davies, and Y. Kivshar, “Nonlinear lightgeneration in topological nanostructures,” NatureNanotechnology , 126–130 (2019).[7] D. Smirnova, S. Kruk, D. Leykam, E. Melik-Gaykazyan, D.-Y. Choi, and Y. Kivshar, “Third-harmonic generation in photonic topological metasur-faces,” Phys. Rev. Lett. , 103901 (2019).[8] S. Mittal, E. A. Goldschmidt, and M. Hafezi, “A topo-logical source of quantum light,” Nature , 502–506(2018).[9] Y. Wang, L.-J. Lang, C. H. Lee, B. Zhang, and Y. D.Chong, “Topologically enhanced harmonic generationin a nonlinear transmission line metamaterial,” Na-ture Communications , 1102 (2019).[10] Y. Zeng, U. Chattopadhyay, B. Zhu, B. Qiang, J. Li,Y. Jin, L. Li, A. G. Davies, E. H. Linfield, B. Zhang,Y. Chong, and Q. J. Wang, “Electrically pumpedtopological laser with valley edge modes,” Nature , 246–250 (2020).[11] Z. Lan, J. W. You, and N. C. Panoiu, “Nonlinearone-way edge-mode interactions for frequency mixingin topological photonic crystals,” Phys. Rev. B ,155422 (2020).[12] T. Ma and G. Shvets, “All-Si valley-Hall photonictopological insulator,” New Journal of Physics ,025012 (2016).[13] X.-D. Chen, F.-L. Zhao, M. Chen, and J.-W. Dong,“Valley-contrasting physics in all-dielectric photonic crystals: Orbital angular momentum and topologicalpropagation,” Phys. Rev. B , 020202 (2017).[14] J.-W. Dong, X.-D. Chen, H. Zhu, Y. Wang, andX. Zhang, “Valley photonic crystals for control of spinand topology,” Nature Materials , 298–302 (2017).[15] X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang,D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surfaceplasmon crystals,” Nature Communications , 1304(2017).[16] J. Noh, S. Huang, K. P. Chen, and M. C. Rechtsman,“Observation of photonic topological valley Hall edgestates,” Phys. Rev. Lett. , 063902 (2018).[17] X. Ni, D. Purtseladze, D. A. Smirnova,A. Slobozhanyuk, A. Al`u, and A. B. Khanikaev,“Spin- and valley-polarized one-way Klein tunnelingin photonic topological insulators,” Science Advances (2018).[18] X.-D. Chen, F.-L. Shi, H. Liu, J.-C. Lu, W.-M. Deng,J.-Y. Dai, Q. Cheng, and J.-W. Dong, “Tunable elec-tromagnetic flow control in valley photonic crystalwaveguides,” Phys. Rev. Applied , 044002 (2018).[19] Y. Kang, X. Ni, X. Cheng, A. B. Khanikaev, and A. Z.Genack, “Pseudo-spin–valley coupled edge states in aphotonic topological insulator,” Nature Communica-tions , 3029 (2018).[20] F. Gao, H. Xue, Z. Yang, K. Lai, Y. Yu, X. Lin,Y. Chong, G. Shvets, and B. Zhang, “Topologicallyprotected refraction of robust kink states in val-ley photonic crystals,” Nature Physics , 140–144(2018).[21] M. I. Shalaev, W. Walasik, A. Tsukernik, Y. Xu,and N. M. Litchinitser, “Robust topologically pro-tected transport in photonic crystals at telecommuni-cation wavelengths,” Nature Nanotechnology , 31–34 (2018).[22] X.-T. He, E.-T. Liang, J.-J. Yuan, H.-Y. Qiu, X.-D.Chen, F.-L. Zhao, and J.-W. Dong, “A silicon-on-insulator slab for topological valley transport,” Na-ture Communications , 872 (2019).[23] D. Smirnova, A. Tripathi, S. Kruk, M.-S. Hwang,H.-R. Kim, H.-G. Park, and Y. Kivshar, “Room-temperature lasing from nanophotonic topologicalcavities,” Light: Science & Applications (2020).[24] Y. Yang, Y. Yamagami, X. Yu, P. Pitchappa, J. Web-ber, B. Zhang, M. Fujita, T. Nagatsuma, andR. Singh, “Terahertz topological photonics for on-chip communication,” Nature Photonics , 446–451(2020).[25] J. Guglielmon and M. C. Rechtsman, “Broadbandtopological slow light through higher momentum-space winding,” Phys. Rev. Lett. , 153904 (2019).[26] E. Sauer, J. P. Vasco, and S. Hughes, “Theory of in-trinsic propagation losses in topological edge statesof planar photonic crystals,” Phys. Rev. Research ,043109 (2020).[27] G. Arregui, J. Gomis-Bresco, C. M. Sotomayor-Torres, and P. D. Garcia, “Quantifying the robust-ness of topological slow light,” Phys. Rev. Lett. ,027403 (2021).[28] D. Anderson and M. Lisak, “Nonlinear asymmetricself-phase modulation and self-steepening of pulses inlong optical waveguides,” Phys. Rev. A , 1393–1398(1983).[29] N. C. Panoiu, X. Liu, and R. M. Osgood, “Self-steepening of ultrashort pulses in silicon photonicnanowires,” Opt. Lett. , 947–949 (2009). [30] J. C. Travers, W. Chang, J. Nold, N. Y. Joly, andP. S. J. Russell, “Ultrafast nonlinear optics in gas-filled hollow-core photonic crystal fibers,” J. Opt. Soc.Am. B , A11–A26 (2011).[31] C. Husko and P. Colman, “Giant anomalous self-steepening in photonic crystal waveguides,” Phys.Rev. A , 013816 (2015).[32] Y. Kivshar and G. Agrawal, Optical Solitons: FromFibers to Photonic Crystals (Elsevier, 2003).[33] M. Soljaˇci´c and J. D. Joannopoulos, “Enhancementof nonlinear effects using photonic crystals,” NatureMaterials , 211–219 (2004).[34] J. M. Dudley, G. Genty, and S. Coen, “Supercontin-uum generation in photonic crystal fiber,” Rev. Mod.Phys. , 1135–1184 (2006).[35] Y. Lumer, Y. Plotnik, M. C. Rechtsman, andM. Segev, “Self-localized states in photonic topologi-cal insulators,” Phys. Rev. Lett. , 243905 (2013).[36] Y. V. Kartashov and D. V. Skryabin, “Modulationalinstability and solitary waves in polariton topologicalinsulators,” Optica , 1228 (2016).[37] D. Leykam and Y. D. Chong, “Edge solitonsin nonlinear-photonic topological insulators,” Phys.Rev. Lett. , 143901 (2016).[38] D. Solnyshkov, O. Bleu, B. Teklu, and G. Malpuech,“Chirality of topological gap solitons in bosonic dimerchains,” Phys. Rev. Lett. , 023901 (2017).[39] W. Zhang, X. Chen, Y. V. Kartashov, V. V. Konotop,and F. Ye, “Coupling of edge states and topologicalBragg solitons,” Phys. Rev. Lett. , 254103 (2019).[40] T. Tuloup, R. W. Bomantara, C. H. Lee, and J. Gong,“Nonlinearity induced topological physics in momen-tum space and real space,” Phys. Rev. B , 115411(2020).[41] R. Chaunsali, H. Xu, J. Yang, P. G. Kevrekidis, andG. Theocharis, “Stability of topological edge statesunder strong nonlinear effects,” Phys. Rev. B ,024106 (2021).[42] M. Guo, S. Xia, N. Wang, D. Song, Z. Chen, andJ. Yang, “Weakly nonlinear topological gap solitonsin Su–Schrieffer–Heeger photonic lattices,” Opt. Lett. , 6466–6469 (2020).[43] S. Xia, D. Juki´c, N. Wang, D. Smirnova, L. Smirnov,L. Tang, D. Song, A. Szameit, D. Leykam, J. Xu,Z. Chen, and H. Buljan, “Nontrivial coupling of lightinto a defect: the interplay of nonlinearity and topol-ogy,” Light: Science & Applications (2020).[44] S. Mukherjee and M. C. Rechtsman, “Observation ofFloquet solitons in a topological bandgap,” Science , 856–859 (2020).[45] L. J. Maczewsky, M. Heinrich, M. Kremer, S. K.Ivanov, M. Ehrhardt, F. Martinez, Y. V. Kartashov,V. V. Konotop, L. Torner, D. Bauer, and A. Sza-meit, “Nonlinearity-induced photonic topological in-sulator,” Science , 701–704 (2020).[46] M. J. Ablowitz, C. W. Curtis, and Y. Zhu, “Localizednonlinear edge states in honeycomb lattices,” Phys.Rev. A , 013850 (2013).[47] M. J. Ablowitz, C. W. Curtis, and Y.-P. Ma, “Linearand nonlinear traveling edge waves in optical honey-comb lattices,” Phys. Rev. A , 023813 (2014).[48] Y. Lumer, M. C. Rechtsman, Y. Plotnik, andM. Segev, “Instability of bosonic topological edgestates in the presence of interactions,” Phys. Rev. A , 021801 (2016).[49] S. K. Ivanov, Y. V. Kartashov, A. Szameit, L. Torner,and V. V. Konotop, “Vector topological edge soli- tons in Floquet insulators,” ACS Photonics , 735–745 (2020).[50] S. K. Ivanov, Y. V. Kartashov, L. J. Maczewsky,A. Szameit, and V. V. Konotop, “Bragg solitons intopological Floquet insulators,” Opt. Lett. , 2271–2274 (2020).[51] J. Cuevas-Maraver, P. G. Kevrekidis, A. Saxena,A. Comech, and R. Lan, “Stability of solitary wavesand vortices in a 2D nonlinear Dirac model,” Phys.Rev. Lett. , 214101 (2016).[52] R. W. Bomantara, W. Zhao, L. Zhou, and J. Gong,“Nonlinear Dirac cones,” Phys. Rev. B , 121406(2017).[53] H. Sakaguchi and B. A. Malomed, “One- and two-dimensional gap solitons in spin-orbit-coupled sys-tems with Zeeman splitting,” Phys. Rev. A , 013607 (2018).[54] A. N. Poddubny and D. A. Smirnova, “Ring Diracsolitons in nonlinear topological systems,” Phys. Rev.A , 013827 (2018).[55] D. A. Smirnova, L. A. Smirnov, D. Leykam, and Y. S.Kivshar, “Topological edge states and gap solitons inthe nonlinear Dirac model,” Laser & Photonics Re-views13