Localized structures in dispersive and doubly resonant optical parametric oscillators
LLocalized structures in dispersive and doubly resonant optical parametric oscillators
P. Parra-Rivas , , L. Gelens , and F. Leo OPERA-photonics, Universit´e libre de Bruxelles,50 Avenue F. D. Roosevelt,CP 194/5, B-1050 Bruxelles, Belgium Laboratory of Dynamics in Biological Systems,KU Leuven Department of Cellular and Molecular Medicine,University of Leuven, B-3000 Leuven, Belgium (Dated: February 26, 2021)We study temporally localized structures in doubly resonant degenerate optical parametric oscil-lators in the absence of temporal walk-off. We focus on states formed through the locking of domainwalls between the zero and a non-zero continuous wave solution. We show that these states undergocollapsed snaking and we characterize their dynamics in the parameter space.
PACS numbers: 42.65.-k, 05.45.Jn, 05.45.Vx, 05.45.Xt, 85.60.-q
I. INTRODUCTION
Localized structures (LSs) can be understood as do-mains of a finite size enclosed by stationary interfaces,and therefore their origin is usually related with the pres-ence of bistability between two steady states [1–3]. In na-ture, they may appear in many different contexts rang-ing from vegetation patches in semi-arid regions or insea grass ecosystems, to localized spots of light in drivennonlinear optical cavities [4–11].LSs are a particular type of so-called dissipative struc-tures that emerge in systems far from the thermodynamicequilibrium due to a self-organization process [12]. Dis-sipative LSs arise due to a double balance between spa-tial coupling and nonlinearity on the one hand, and gainand dissipation on the other hand [13]. Spatial couplingappears for example through the dispersion and/or dis-persion of the light in optical systems, and is associatedwith diffusion processes in chemistry, biology and ecology[14, 15].In optics, dissipative LSs have been widely studied inthe context of externally driven diffractive nonlinear cav-ities with either cubic χ (3) (Kerr) [16, 17] or quadratic χ (2) nonlinear media [18–24]. In these cavities, LSs formin the plane transverse to the propagation direction, andthey are commonly known as spatial cavity solitons. LSshave been also studied in wave-guided dispersive Kerrcavities, where LSs correspond to temporal pulses aris-ing along the propagation direction, and they are one-dimensional [11, 25–27]. Temporal LSs have been con-sidered as the basis for all-optical buffering [11], and inthe last decade, also for the generation of broadband fre-quency combs in microresonators [28–30].Recently, it has been shown that dispersive cavitieswith quadratic nonlinearities may provide an alternativeto Kerr cavities for the generation of frequency combs[31–35]. In contrast to Kerr combs, quadratic ones mayoperate with decreased pump power and can reach spec-tral regions that were not accessible before. Therefore,understanding the formation of temporal LSs is impor- FIG. 1: (Color online) Schematic example of a doublyresonant DOPO. A ring resonator with a χ (2) nonlinearity is driven by a CW field B in at frequency2 f . The quadratic interaction gives rise to a field A with carrier frequency f that resonates together with B , and therefore to two frequency spectra around 2 f and f , respectively.tant in this context.In this work we study the formation of LSs throughthe locking of domain walls (DWs) in a χ (2) -dispersivecavity matched for degenerate optical parametric oscil-lations (DOPO). A schematic example of such type ofcavity is shown in Fig. 1. The cavity is externally drivenby a pump field B in at frequency 2 f , and a field A is gen-erated at frequency f through parametric down conver-sion. We consider a doubly resonant configuration suchthat both fields A and B resonate together in the cav-ity. In such systems, continuous-wave (CW) states maycoexist for the same values of a control parameter (bista-bility), and DWs connecting them can eventually form.DWs, also known as wave fronts or switching waves, ex-hibit a particle-like behavior in such a way that they caninteract and lock, thus forming LSs of different extensions[1, 2].DWs have been previously studied in the context ofdiffractive DOPOs [23, 36], and the formation of LSsthrough their locking has been analyzed in detail for bothsingly and doubly resonant configurations [22, 23]. Re-cently, the formation of LSs has also been studied in dis-persive DOPOs and in the presence of temporal walk-off[37].In all these studies, DWs and LSs form between CW a r X i v : . [ n li n . PS ] F e b states that have the same amplitude and are equally sta-ble. As such they are also called equivalent CW solutions.However, in DOPOs, bistability between non-equivalentCW states is also present, and DWs and LSs may ariseas well. Nonetheless, as far as we know, the formation ofthis type of LSs has not been analyzed in detail, neitherin diffractive nor in dispersive cavities. Hence, in thispaper we elucidate the formation, dynamics and bifurca-tion structure of the last type of LSs (hereafter type-I)and their connection with the former LSs (type-II). Inthis work we neglect the effect of the temporal walk-off.The manuscript is organized as follows. In Sec. II weintroduce the mean-field model describing doubly reso-nant dispersive DOPOs and derive a single model witha nonlocal nonlinearity. In Section III we present thestationary problem, analyze the CW solutions and theirlinear stability, and introduce the locking of DWs as themechanism behind the formation of LSs. Later, in Sec. IVwe calculate, applying multi-scale perturbation methods,a weakly nonlinear pulse-like solution about the trivialCW state. From Secs. V to VII, we then study the bi-furcation structure of the different types of LSs formedthrough the locking of DWs, and how this structure ismodified when varying the control parameters of the sys-tem. Finally, in Sec. VIII, we discuss the main results ofthe paper. II. MEAN-FIELD MODELS
In this section we introduce the mean-field model for adispersive DOPO in a doubly resonant configuration andwe derive a nonlinear nonlocal model that will be usedin the remainder of this work.Assuming that the resonator exhibits high finesse, thatboth fields do not vary significantly over a single round-trip (i.e., the combined effects of nonlinearity and disper-sion are weak), and following Refs. [32, 38], the dynamicsof a DOPO can be described by a mean-field model forthe slowly varying envelopes of the signal electric field A centered at frequency ω and, the pump field B centeredat the frequency 2 ω , as already shown in Ref. [37]. Thenormalized mean-field model reads: ∂ t A = − (1 + i ∆ ) A − iη ∂ x A + iB ¯ A (1a) ∂ t B = − ( α + i ∆ ) B − (cid:0) d∂ x + iη ∂ x (cid:1) B + iA + S. (1b)In the current formulation, t corresponds to the normal-ized slow time describing the evolution of fields after ev-ery round-trip at a fixed position in the cavity, and x isthe normalized fast time [37]. The parameter α is theratio of the round-trip losses α , associated with thepropagation of the signal and pump fields, ∆ , are thenormalized cavity phase detunings, η , and the groupvelocity dispersion (GVD) parameters of A and B , d isthe normalized rate of temporal walk-off or wavevectormismatch related with the difference of group velocities between both fields, and S is the driven field amplitudeor pump at frequency 2 ω . With the normalization usedhere η = +1( −
1) denotes normal (anomalous) GVD,and η can take any value positive or negative.The system of equations (1) are formally equivalent tothose describing diffractive spatial cavities [39, 40]. Inthat context, η ≈ η with η j > x represents a transverse spatial dimension,and ∂ x applied to either A and B the beam diffraction.In contrast to spatial cavities, where the walk-off is nor-mally negligible, in dispersive cavities it is very large andshould be taken into consideration. The walk-off imposessevere restrictions on the efficiency of optical parametricamplification and often prevents the formation of LSs.Hence, it would be desirable to suppress it. This can bedone by dispersion engineering as already shown in [35].Thus, in the following we will consider d = 0. The effectsof the walk-off on the stability and dynamics of LSs isbeyond the scope of the present paper, and will be pre-sented elsewhere. Furthermore, we will consider perfectphase-matching, what in wave-guided systems, as the onediscussed here, implies ∆ = 2∆ .The numerical exploration of the dynamics of Eqs. (1)for a large range of parameters suggests that the B fieldvaries slowly in t . Thus, assuming that ∂ t B ≈
0, and fol-lowing Refs. [32, 37, 41], we can further simplify Eqs. (1)to a single mean-field model for A [see Appendix A] witha nonlocal nonlinearity: ∂ t A = − (1 + i ∆ ) A − iη ∂ x A − ¯ A ( A ⊗ J ) + ρ ¯ A , (2)where ⊗ denotes convolution with the nonlocal kernel J ( x ) = 1 + ˜∆ π (cid:90) ∞−∞ e − ikx dk i ( ˜∆ − ˜ η k ) , (3)with ˜∆ = ∆ /α , ˜ η = η /α , although in the followingwe drop (˜ · ).The normalized field reads A = Ae − iψ (cid:112) α (1 + ∆ ) (4)with ψ = π/ − ∆ ) / , (5)and the normalized pump amplitude ρ = Sα (cid:112) . (6)Equation (2) is a kind of parametrically forced Ginzburg-Landau (PFGL) equation [42] with a long range couplingin x introduced by the nonlocal nonlinearity A ⊗ J . Inthis framework the interaction between A and B is equiv-alent to the propagation of A in a medium with a nonlocalnonlinearity leading to an effective third order nonlinear-ity.With this approximation, the B field is dynamicallyslaved to A , and explicitly given by B = ( − A ⊗ J + ρ ) e i atan( − ∆ ) . (7)Models with a similar type of nonlocal response have al-ready been considered in single-pass problems [41] andin quadratic dispersive cavities [31–34]. In particularEq. (2) is formally equivalent to the mean field modelderived in [33, 34] for the description of a singly resonantDOPO (with a different response function).In all these cases the nonlocal response in Eq. (2) de-pends on A , in contrast to other nonlocl models de-scribing Raman [43, 44], diffusion [45, 46] or thermal[45, 47, 48] effects, where the nonlocal response dependson the intensity | A | .The models (1) and (2) are equivalent when studyingstationary states, such as LSs. Unless stated otherwise,here we focus on the study of Eq. (2).In terms of the real and imaginary part of A = U + iV Eq. (2) yields the system ∂ t (cid:20) UV (cid:21) = ( L + N ) (cid:20) UV (cid:21) , (8)with L and N being the linear and nonlinear operatorsdefined by L = (cid:20) ρ − + η ∂ xx − ∆ − η ∂ xx − ( ρ + 1) (cid:21) (9)and N = − (cid:20) N a N b N b −N a (cid:21) , (10)with coefficients N a = U ⊗ J R − V ⊗ J R − U V ⊗ J I , (11a) N b = U ⊗ J I − V ⊗ J I + 2 U V ⊗ J R , (11b)where J R and J I correspond to the real and imaginaryparts of the Kernel J [see Appendix A]. In the followingwe focus on the normal GVD regime ( η = +1), andchoose α = 1. III. STATIONARY SOLUTIONS
In this work we focus on the study of stationary states.In the current mean-field formulation these states sat-isfy ( ∂ t A, ∂ t B ) = (0 , − iη A xx − (1 + i ∆ ) A − ¯ A ( A ⊗ J ) + ρ ¯ A = 0 , (12)or, equivalently, stationary states are solutions of( L + N ) (cid:20) UV (cid:21) = (cid:20) (cid:21) . (13) Stationary states can be of different nature such as ho-mogeneous CW states [49], periodic patterns [39, 50], orDWs and LSs [23, 36]. Notice that equations (2) and (12)are invariant under the transformations x → − x , and A → − A . The first symmetry means that any station-ary solution is left/right symmetric (i.e. has a reflectionsymmetry), and according to the second symmetry, if A s is a solution, so is − A s .As stated before, in this work we focus on the studyof LSs formed through the locking of DWs connectingdifferent CWs. Hence, in this section we introduce theCW solutions of Eq. (12), analyze their linear stability,and study the formation of LSs. A. Continuous wave solutions
The CW states of this system were first studied inRef. [49] in the context of diffractive cavities. Here wereview some of the results of that study in terms of thenonlinear nonlocal model (2). In this framework the CWscorrespond to the homogeneous steady state solutions ofEq. (2), which satisfy the algebraic equation: − (1 + i ∆ ) A s − (1 − i ∆ ) A s | A s | + ρ ¯ A s = 0 . (14)Writing A s = | A s | e iφ , Eq. (14) becomes (cid:2) − (1 + i ∆ ) − (1 − i ∆ ) | A s | + ρe − iφ (cid:3) | A s | = 0 , (15)which yields three solutions: the trivial state A s = A =0, and the two non-trivial states A ± = | A ± | e iφ ± , with | A ± | = (∆ ∆ − ± (cid:112) (1 + ∆ ) ρ − (∆ + ∆ ) , (16)and phase φ ± = acos (cid:2) ( | A ± | + 1) /ρ (cid:3) / . (17)If ∆ ∆ >
1, only the A + branch exists, and bifurcatessuper-critically from a pitchfork bifurcation [51] occur-ring at pump strength ρ a = (cid:113) . (18)The pitchfork bifurcation defines a line in the phase dia-gram in the (∆ , ρ ) − parameter space plotted in Fig. 2(a)[see solid black line]. An example of the HSS bifurcationdiagram in the super-critical regime is shown in Fig. 2(b)for ∆ = − .
5. In contrast, for ∆ ∆ < A − arisessub-critically as shown in Fig. 2(c) for ∆ = −
2, andundergoes a fold or turning point [51] at ρ t = ∆ + ∆ (cid:112) , (19)where it merges with A + . This line is plotted in greenin Fig. 2(a). The transition between these two regimesFIG. 2: (Color online) In (a) the phase diagram in the(∆ , ρ ) − parameter space showing the principalbifurcation lines of the CW solutions: the pitchforkbifurcation ρ a (black line), and the fold or turning line ρ t corresponding to SN t (green line). (b) shows the CWsolutions for ∆ = − .
5, and (c) those for ∆ = − ∆ = 1, orequivalently ∆ = 1 / √ • Region I: Only A exists and is stable. This regionis spanned by the parameter region below ρ a for∆ < / √
2, and ρ < ρ t for ∆ > / √ • Region II: The non-trivial solution A + coexists with A that is now unstable. This region is spanned by ρ > ρ a . • Region III: Solutions A , A − and A + coexist, where A and A + are both stable. This region is spannedby the values of ρ such that ρ t < ρ < ρ a . B. Linear stability analysis of the continuous wavesolutions
Here we perform the linear stability analysis of theCW solutions in the presence of dispersion. Dispersioncan cause the emergence of pattern forming instabilities,such as the Turing or modulational instability (MI) [52].In the absence of dispersion, it is known that A can un-dergo a Hopf instability leading to self oscillations, period doubling and chaos [49]. Later the analysis was extendedto include the effect of diffraction in the context of spatialcavities [21], and the spatio-temporal dynamics arisingfrom the interaction of the Turing and Hopf modes wasexamined in detail in Refs. [53, 54]. In this work we focuson the bistable regime (∆ ∆ < A ( t, x ) = A s + (cid:15)ζe σt + ikx + c.c., (20)describing a small modulation about the CW A s , where σ is the growth rate of the perturbation, and ζ the eigen-vector associated with the linearization of Eq. (2) at or-der (cid:15) . The linear problem has modulated solutions if thegrowth rate satisfies σ + a σ + a = 0 , (21)where a = 2(1 + 2 I s F [ J R ]) , (22a) a = c I s + c I s + c , (22b)and c = 4 (cid:0) F [ J R ] − ( η k − ∆ ) F [ J I ] (cid:1) (23a) c = 4( F [ J R ] + F [ J I ] ) , (23b) c = η k − η ∆ k . (23c)Here F denotes the Fourier transform as defined in Ap-pendix A. The CW solutions A and A ± are linearly sta-ble to perturbations with a given k if Re[ σ ( k )] <
0, andunstable otherwise. When k = 0 we recover the homoge-neous stability analysis performed in Ref. [49], howeverwhen k is allowed to vary the system can undergo a MIand periodic patterns may appear.Through a linear stability analysis of the trivial solu-tions A s = A , we obtain that A undergoes a MI at ρ = ρ c ≡ , (24)where patterns with a characteristic wavenumber k c = (cid:112) η ∆ , (25)arise, provided that η ∆ >
0. Two situations can bedistinguished depending on the sign of the product η ∆ .When η = 1 (normal GVD regime), A undergoes a MI if∆ >
0. In contrast, when η = − <
0. Notice that the stability of thetrivial state does not depend on η .FIG. 3: (Color online) Panels (a)-(b) show the marginalinstability curve and the bifurcation diagram associatedwith the CW solution for (∆ , η ) = ( − , − . I s where the CW isunstable, and correspond to the dashed lines plotted in(b). The CW solution is stable outside this region asshown with solid lines in (a). Panels (c)-(d) show thesame type of diagrams but for (∆ , η ) = ( − , − . ρ M of thesystem for such values of the parameters.The linear stability analysis of the non-trivial CWstates A ± is cumbersome and an exact analytical expres-sion of the MI threshold and critical wavenumber do notexist [21]. Nevertheless, we can analyze the stability ofthese states by means of the marginal instability curve I s ( k ). This curve defines the band of unstable modes,and is composed by two branches I ± s ( k ) satisfying thequadratic equation obtained by setting σ = 0 in Eq. (21): c I s + c I s + c = 0 . (26)The CW state is unstable against a perturbation with afixed k , if I s is inside the curve, i.e. I s ( k ) − < I s < I + s ( k ),and unstable otherwise. For k (cid:54) = 0 the extrema ( k, I s ) =( k c , I c ) of this curve define the MI.Figure 3 (a) shows the marginal instability curve as-sociated with the CW solution shown in panel (b) for(∆ , η ) = ( − , − . I t for k = 0, and therefore A − is unstable from ρ a to SN t [see dotted line in Fig. 3(b)], while A + is stablefor any value of k (cid:54) = 0 as shown in Fig. 3(b).Decreasing the value of | η | the maximum migratesfrom the fold SN t at ( k, I s ) = (0 , I t ) to ( k, I s ) = ( k c , I c )where a MI takes place. This is the situation shown inFig. 3(c) for η = − .
05. In this case A − remains unsta-ble, and A + is stable above the MI, i.e. for I s > I c , and unstable otherwise [see solid and dotted lines inFig. 3(d)].The MI defines a manifold ρ c = ρ ( I s ( k c ) , η , ∆ ) ac-cording to which region III can be subdivided as follows: • III a : A + is unstable in response to non-homogeneous perturbations (i.e. k (cid:54) = 0). This re-gion spans the parameter space ρ t < ρ < ρ c . • III b : A + is stable in response to non-homogeneousperturbations. This region spans the parameter re-gion ρ > ρ c . C. Formation of localized states through domainwall locking
As shown in the previous sections, the CW solutionsmay coexist stably depending on the range of parame-ters. Therefore, in the presence of dispersion, DWs mayarise connecting two different CWs. In this context twodifferent types of DWs occur: • Type-I: the connection occurs between A and A + [see Fig. 4(a), left]. They exist in region III b . • Type-II: the connection arises between two equiv-alent (equally stable) non-trivial states, i.e. − A + and A + . They occur in regions II and III b . [seeFig. 4(b), left]The tails of both type-I and type-II DWs around the CW-solution A + [see close-up view in (a)] can be describedasymptotically by the ansatz A ( x ) = A + s + (cid:15)e λx + c.c. ,where the eigenvalues λ satisfy the condition σ ( − iλ ) = 0,and are therefore solutions of the polynomial b λ + b λ + b λ + b λ + b = 0 , (27)where the coefficients b m are functions of the parametersof the system.Due to the reflection symmetry x → − x , Eq. (27) isinvariant under λ → − λ , and λ → ¯ λ [55]. Equation (27)cannot be solved analytically except in some particularconditions [22]. The tails can approach A + either mono-tonically, or in a damped oscillatory fashion. The lattercase is related with the existence of at least four complexeigenvalues λ , , , = ± Q ± iK , those with the smallestreal part | Q | . The oscillatory damped tails are describedby A ( x ) = A + s + a cos( Kx ) e − Qx . (28)In contrast, when K = 0 the oscillations disappear, andthe DW approaches A + monotonically. In what followswe separately describe the interaction of DWs and theformation of type-I and type-II LSs . tangency FIG. 4: (Color online) (a,b) The real component of a DW of type-I ( A → A + ) (a) and a DW of type-II( − A + → A + ) (b). (c) Sketch of the oscillatory interaction defined by Eq. (29) at the Maxwell point ( ν = 0) and twolocations away from the Maxwell point (i.e. ν = ν and ν ). The stable (unstable) separations D s are labeled using • ( ◦ ); (d,e) Example of Type I (d) and Type II (e) LSs. Type-I domain walls and localized structures
The CW states A and A + are non-equivalent, andtype-I DWs move with a constant velocity that dependson the control parameters of the system. In gradientsystems, where an energy functional can be defined, thevelocity is proportional to the energy difference between A and A + . In this context, the Maxwell point of thesystem is defined as the parameter value where both CWstates have the same energy, or equivalently, as the pointwhere the velocity of the DWs becomes zero [56]. Here,despite the system not having gradient dynamics, we willstill refer to such a point as the Maxwell point, and here-after we label it as ρ M . This point is marked using adotted-dashed line in Figs. 3(b) and (d). In a range ofparameters around ρ M two DWs with different polarity,let say a kink A → A + and anti-kink A + → A , sepa-rated by a distance D interact as described by ∂ t D = (cid:37) cos( KD ) e − QD + ν ≡ f ( D ) , (29)where ν ∼ ρ − ρ M , measures the distance from theMaxwell point ρ M , and (cid:37) depends on the parameters ofthe system [2].When K (cid:54) = 0, the oscillatory nature of the interactionleads to alternating regions of attraction and repulsion[see Fig. 4(c)]. DWs lock at different stationary sepa-rations D s satisfying f ( D s ) = 0. At ρ = ρ M ( ν = 0)[Fig. 4(c), bottom], the width of the LSs ( D s ) is quan-tized: D ns = π K (2 n + 1), with n = 0 , , , . . . [1, 2].Figure 4(d) shows an example of a LS of width D s , corre-sponding to the stationary distances shown in Fig. 4(c).The stable (unstable) separation distances are markedwith • ( ◦ ). We refer to these states as Type-I LSs. When ν (cid:54) = 0 the red curve shifts upwards or downwards (de-pending on the sign of ν ), and as a result, the number ofstationary intersections decreases as is shown in Fig. 4(c)for ν = ν and ν . Hence, when moving away from the Maxwell point ρ M , the widest LSs disappear first, buteventually even the single peak LS is lost. In Sec. V wewill see that the interaction described by Eq. (29) is re-sponsible of the bifurcation structure that the previousLSs undergo.When the tails are monotonic ( K = 0) a different phe-nomenon known as coarsening occurs where two DWswith different polarity attract each other until eventuallythey annihilate one another [57]. Type-II domain walls and localized structures
In regions II and III b the solutions − A + and A + co-exist and are linearly stable, and hence type-II DWsconnecting them may also arise. In this case theseCWs are equivalent, and therefore, the DWs are station-ary [see Fig. 4(b)]. Here the interaction between kink( − A + → A + ) and anti-kink ( A + → − A + ) is describedby Eq. (29) by setting ν = 0 [1] (see Fig. 4(c)). The LSsresulting from this interaction are referred to as type-II LSs, and have been largely studied in the context ofdiffractive cavities [22, 23]. DWs of this type may un-dergo non-equilibrium Ising-Bloch transition, where DWsstart to drift [58], and as result LSs may show very com-plex dynamics [59]. An example a such type of state isshown in Fig. 4(e). IV. WEAKLY-NONLINEAR SOLUTIONSAROUND THE PITCHFORK BIFURCATION
While the locking of DWs explains the formation ofhigh amplitude LSs, it does not describe their originfrom a bifurcation point of view. In this section we showthat those structures are connected with small ampli-tude states that arise from the Pitchfork bifurcation oc-curring at ρ a . In order to do so we derive a stationarynormal form for the pitchfork bifurcation by applyingweakly nonlinear multi-scale analysis. We find two typesof extended solutions that explain the origin of the struc-tures discussed in Sec. III. The solutions of this normalform have been studied in the context of the paramet-rically forced Ginzburg-Landau equation [42]. However,in our case, we have a long-range nonlocal coupling in x in terms of the nonlocal nonlinearity A ⊗ J . In order todeal with this difficulty we follow the approach shown inRef. [60].Following [42] we fix ∆ and consider the asymptoticexpansion of the fields U , and V as a function of theexpansion parameter (cid:15) defined by ρ = ρ a + δ(cid:15) , where δ is the bifurcation parameter. Then the expansion reads (cid:20) UV (cid:21) = (cid:15) (cid:20) u v (cid:21) + (cid:15) (cid:20) u v (cid:21) + · · · . (30)where we allow each of the terms in the previous expan-sion to depend just on the long scale x ≡ (cid:15)x . Consider-ing Eq. (30) the linear operator expands as L = L + (cid:15) L , (31)with L = (cid:20) ρ a − − ∆ − ( ρ a + 1) (cid:21) , (32a)and L = (cid:20) δ η ∂ x − η ∂ x − δ (cid:21) . (32b)Similarly the nonlinear operator becomes N = (cid:15) N = − (cid:20) N a N b N b −N a (cid:21) , (33)with N a = u ⊗ J R − v ⊗ J R − u v ⊗ J I (34a) N b = u ⊗ J I − v ⊗ J I + 2 u v ⊗ J R . (34b)The insertion of the previous expansions in the stationaryequation (13) yields a hierarchy of equations for succes-sive orders in (cid:15) , which up to third order read: O ( (cid:15) ) : L (cid:20) u v (cid:21) = (cid:20) (cid:21) , (35a)and O ( (cid:15) ) : L (cid:20) u v (cid:21) + ( L + N ) (cid:20) u v (cid:21) = (cid:20) (cid:21) (35b)At first order in (cid:15) the solvability condition provides, ρ a = (cid:113) ∆ + 1 , (36) FIG. 5: (Color online) Weakly nonlinear solutionaround the pitchfork bifurcation ρ a . (a)-(b) show inblue the real and imaginary profiles of the weaklynonlinear state given by (44) for ∆ = − ρ − ρ a = 0 .
01. Red dashed lines represent the numericalsolutions of Eq. (12) at the same point. Both lines areindistinguishable.which confirms the position of the pitchfork bifurcationalready calculated in Sec. III. The solutions at this orderare of the form (cid:20) u v (cid:21) = (cid:20) ξ (cid:21) a ( x ) , (37)where ξ = ∆ / (1 − ρ a ) and a ( x ) is the real envelope am-plitude to be determined at next order in the expansion.Applying the same procedure as in Ref. [60] we show [seeAppendix B] that the the solvability condition at O ( (cid:15) )gives the stationary normal form for the amplitude a : C ∂ x a = δa + C a , (38)with the coefficients C = − η ξ ξ (39a)and C = 1 − ξ ( ξ + 2∆ ) (39b)This last equation admits the CW solutions a = (cid:112) − δ/C , or equivalently, (cid:20) UV (cid:21) = ∆ − ρ a (cid:114) ρ − ρ a − C + · · · , (40)that confirms the result already obtained in Sec. III: theCW bifurcates super-critically ( ρ > ρ a ) if ∆ ∆ >
1, andsub-critically ( ρ < ρ a ) if ∆ ∆ < ∆ >
1) the normalform (38) admits DW-like solutions of the form a ( x ) = (cid:114) δ − C tanh (cid:32)(cid:114) δ − C x (cid:33) , (41)yielding a super-critical bifurcation to states of the form (cid:20) UV (cid:21) = ∆ − ρ a (cid:114) ρ − ρ a − C tanh (cid:18)(cid:114) ρ − ρ a − C x (cid:19) + · · · (42)This analytical solution was first obtained in the contextof diffractive cavities in Ref. [21], where the normal formaround ρ a was derived in terms of the full model (1).In contrast, for ∆ ∆ <
1, the normal form (38) ad-mits solutions of the form a ( x ) = (cid:114) − δC sech (cid:32)(cid:114) δC x (cid:33) , (43)which provides the sub-critical emergence of type-I LSs (cid:20) UV (cid:21) = ∆ − ρ a (cid:115) ρ a − ρ ) C sech (cid:18)(cid:114) ρ a − ρ − C x (cid:19) + · · · , (44)These weakly-nonlinear solutions are only valid close tothe pitchfork bifurcation at ρ a . In the next section weshow how these solutions are modified when entering thehighly nonlinear regime as one of the control parame-ters of the system is changed. Notice that the weaklynonlinear solutions (42) and (44) are independent of theparameter η . This shows that in the weakly nonlinearregime the states studied here are not influenced by thepresence of the long-range interaction in x .In the coming section we focus on the sub-criticalregime and study the bifurcation structure of LSs of theform (44). To check the validity of our calculations, inFig. 5 we have plotted the real and imaginary parts (blueline) of the weakly nonlinear state (44) together with thenumerical solutions (dashed red line) obtained through aNewton-Raphson solver, showing excellent agreement. V. BIFURCATION STRUCTURE OF TYPE-ILOCALIZED STATES
In this section we study the bifurcation structure of thetype-I LSs. In Sec. IV we have derived a normal formequation around the pitchfork bifurcation occurring at ρ a . Two stationary weakly nonlinear solutions are foundcorresponding to a small amplitude DW and bump [seeEq. (42) and Eq. (44)] that arise in the super-critical andsub-critical regime, respectively.In what follows we focus on the sub-critical regimeand, unless stated otherwise, fix ∆ = −
2, and η < ρ a . However, applying numeri-cal continuation techniques [61] we are able to track thesesolutions to parameter values away from the small am-plitude bifurcation ρ a , and therefore, to build bifurcationdiagrams as those shown in Fig. 6. In these diagrams the L -norm || A || = L (cid:82) L/ − L/ | A ( x ) | dx is plotted as a func-tion of the pump intensity ρ for different values of η .Figures 6(a),(d) show the bifurcation diagram for η = − .
8, where panel (d) is a close-up view of the diagramshown in panel (a). The blue lines in Fig. 6(a) representthe CW solution, whose linear stability is shown usingsolid (dashed) lines for stable (unstable) solutions. The vertical gray line corresponds to the Maxwell point ofthe system ρ M . At this point the velocity of the DWsconnecting the trivial solution A with the non-trivial one A + is zero, and around this point two DWs of differentpolarities can lock to each other and form LSs of type-I,as already discussed in Sec. III. Close to ρ a the LS iswell described by the small amplitude weakly nonlinearsolution (44), and is initially unstable.The stability of the x − dependent steady states is ob-tained from the analysis of the eigenspectrum of the lin-ear operator associated with Eq. (2) evaluated at suchsteady state. This linear operator must be calculatednumerically, and hence, it corresponds to the Jacobianmatrix associated with the coupled algebraic equationsthat originate from discretizing Eq. (2). To confirm thevalidity of the stability results we have also performedsuch analysis using the full model (1). Indeed, for thetype of states studied here, the stability analysis usingboth models agrees.Decreasing ρ the amplitude of the LSs increases [seeprofile (i)] until reaching the first fold of the diagram.This fold correspond to a saddle-node bifurcation that welabel as SN l [see Fig. 6(d)]. Once SN l is passed the LSbecome stable. At this stage the LS corresponds to a highamplitude state as the one shown in panel (ii). Increasing ρ further the amplitude of the LS grows, and it becomesunstable at a second saddle-node SN r [see inset]. At thesame time a small dip is nucleated in the central positionof the LS forming an almost flat plateau [see panel (iii)].While increasing the norm the LS broadens and becomesstable one more time at SN l [see profile (iv)]. Proceed-ing up in the diagram (i.e. increasing || A || ) the processrepeats, resulting in the broadening of the LSs as shownin panel (v). At this stage one can observe how the LSis formed by a pair of DWs connecting A with A + ofdifferent polarities, namely DWs + and DWs − .In the course of this process the solution branches un-dergo a sequence of exponentially decaying oscillationsin ρ at the vicinity of the Maxwell point ρ M ≈ . collapsed snaking [42, 62, 63], and has beenstudied in detail in the context of Kerr cavities [64].In periodic systems like ours the LS branch moves awayfrom ρ ≈ ρ M when the maximum amplitude starts todecrease below A − and the solution turns into a darkLS sitting on A + [see profile (vi) translated L/ t , where the amplitude of theLS becomes zero. In terms of spatial dynamics this pointcorresponds to a reversible Takens-Bogdanov bifurcation[51, 55], and a weakly nonlinear solution of the form A − A + ∼ a sech ( bx ) can be obtained as already donein Refs. [64–66].The bifurcation diagrams shown in Fig. 6(a) and (d)correspond to a slice for constant η = − . η , ρ ) − parameter space for constant ∆ = −
2. Thesaddle-node and the pitchfork bifurcations of the CW (v)(vi)SN t A + A − A (a) (i)(iii)(iv)(v)SN l (d) (iii)(iv)SN l SN r SN t MISN l (b) SN l SN r SN l SN r (e) (vii)(viii)(ix)(x) (xi)(xii) SN t MI(c) (xvii)(xviii) (xiii)(xiv)(xv)(xvi)SN l SN r SN l SN r (f)
15 0 1501 U , V (i)
15 0 1501 (ii)
15 0 1501 (iii)
15 0 1501 (iv)
25 0 2501 DW + DW − (v)
25 0 2501 (vi)
15 0 1501 U , V (vii)
15 0 1501 (viii)
15 0 1501 (ix)
15 0 1501 (x)
15 0 1501 (xi)
15 0 1501 (xii)
15 0 15 x U , V (xiii)
15 0 15 x (xiv)
15 0 15 x (xv)
15 0 15 x (xvi)
25 0 25 x (xvii)
25 0 25 x (xviii) FIG. 6: (Color online) Bifurcation diagrams for LSs of type-I at ∆ = − η . In (a),(d)collapsed snaking for η = − .
8, in panels (b),(e) for η = − .
2, and (d),(f) correspond to η = − .
05. The panels(d), (e), and (f) are close-up views of the bottom parts of the bifurcation diagrams shown in (a), (b), and (c). Solid(dashed) lines correspond to stable (unstable) solutions. The vertical gray point-dashed line stands for ρ M , and redand orange vertical lines in panel (f) refer to Fig. 4(c). The different SNs of the LSs are labeled through SN l,ri , andthe red dots correspond to the LSs shown in the subpanels (i)-(xviii), where blue and green solid lines represent U and V respectively. ρ t and ρ a are plotted in black and green solid lines re-spectively. The Maxwell point ρ M is indicated with a redsolid line, the MI ρ c is shown in purple, and the SN l andSN r are plotted in blue. The gray area in-between theselines is the region where type-I LSs exist. Increasing η the different folds SN li and SN ri with i = 1 , , ... approach one another and disappear in a sequence of cusp bifur-cations. Here we only show the cusp that involves thecollision of SN l and SN r .When decreasing | η | , the situation is rather different.The MI instability, not present before, arises from SN t around η ≈ − . η , ρ ) − parameter space for ∆ = −
4. The gray arealimited by SN r and SN l corresponds to the parameterregion where LSs of type I exist. The red and purplelines correspond to the Maxwell point ρ M and the MI ρ c respectively. The horizontal bifurcation lines in blackand green are the pitchfork bifurcation ρ a and thesaddle-node bifurcation ρ t of the CW solution. Theinset shows a close-up view about the cusp bifurcation(C) where SN r and SN l collide and disappear. Thepointed, dashed-pointed, and dashed vertical linescorrespond to the bifurcation diagrams shown in Fig. 6for ∆ = − . , − .
2, and − . η , destabilizing the CW branch A + . Figure 6(b),(e) shows the bifurcation diagram cor-responding to this situation for η = − .
2. As in theprevious case, a branch of LSs arises from the pitchforkbifurcation at ρ a and undergoes collapsed snaking. How-ever, in this case, the Maxwell point, and the bifurcationdiagram itself have shifted to higher values of ρ . Fur-thermore, while in Fig. 6(a)-(d) the solutions branchescollapse rapidly to ρ M as increasing the || A || , in panels(b)-(e) the collapse is much slower, and hence the solu-tion branches of wider structures persist.The profiles (vii-xii) show how the LSs are modi-fied while passing through two consecutive folds [seeFig. 6(e)]. In (i) the the LS consist in a single bump.Soon after passing SN r the structure start to develop acentral dip [see profile (vii)] that deepens as decreasing ρ until reaching SN l [see (viii)] where it becomes stable.This process repeats: at every SN ri a new dip is nucleatedfrom the center of the LS which broadens as increasing || A || [see profiles (viii)-(xii)].As before, the branch of LSs detaches from ρ M ≈ . || A || ≈ .
6, and persists until it meetswith A + . Here, however, the merging occurs not atthe SN t , but at the MI at ρ c ≈ . π/k c exist and arise sub-critically from ρ c together with a bump solutions of theform A − A + ∼ a sech( bx )cos( k c x + ϕ ), where a and b depend on the control parameters of the system, and ϕ controls the phase of the pattern within the sech [66, 67].This structure is plotted in panel (xviii) of Fig. 6 for η = − .
05. These type of LSs may undergo homoclinicsnaking [68, 69], although for the range of parametersexplored here, such type of structure has not been found.The collapsed snaking structure is a consequence ofthe damped oscillatory interaction between the two DWsforming the LSs of type I (see Sec. III). To understandthis phenomenon let us take a look to the sketch shownin Fig. 4(c). At the Maxwell point ( ν = 0) a numberstable and unstable LSs form at the stationary DWs sep-arations ∆ ns . The stable (unstable) LSs in Fig. 4(c) thencorrespond to a set of points on top of the stable (unsta-ble) branches of solutions at ρ M in the collapsed snakingdiagrams of Fig. 6 [see for example the diagram shownin panel (f)]. As ρ moves away from ρ M , the branches ofwider LSs start to disappear in a sequence of SN bifur-cations, and only narrow LSs survive. At this point [seered vertical line in Fig. 6(f)] the scenario corresponds tothe situation shown in Fig. 4(c) for ν = ν , where fourintersections of f ( D ) with zero take place. Decreasing ρ even further only two intersections occur [see Fig. 4(c)for ν = ν ] which correspond to the stable and unstablesingle peak branches [see orange vertical line in Fig. 6(f)].In this context, the SN bifurcations of the collapsedsnaking diagram take place when the extrema of f ( D )become tangent to zero. Indeed, the tangency observedin Fig. 4(c) corresponds to the occurrence of SN l . Even-tually the last tangency corresponding to SN l occurs andthe single peak LS is destroyed.Decreasing | η | to even lower values, the morphologyof the collapsed snaking does not change much, despitethe widening of the solution branches.As a result, the theregion of existence of the LSs increases [see Fig. 7]. Thisis the situation shown in Fig. 6(c)-(f) for η = − . η is related with the modification of the oscil-latory tails of the DWs involved in the formation of theLSs. It therefore depends directly on the spatial eigen-values λ . Indeed, decreasing | η | the oscillations in thetails become less damped, and its wavelength shortens.This can be appreciated when comparing the LSs plottedin panels (ii)-(vi) with those shown in (xiv)-(xviii).The limit η → η = 0 the nonlocal nonlinear term becomes A ⊗ J =(1 − i ∆ ) A , and Eq. (12) reduces to ∂ t A = − (1 + i ∆ ) A − iη ∂ x A − (1 − i ∆ ) | A | A + ρ ¯ A , (45)which is a particular version of the more general paramet-rically forced Ginzburg-Landau (PFGL) equation with2:1 resonance, which has been studied in detail in [42].We have confirmed, although not shown here, that thesame type of solutions reported in this work are alsopresent in model (45). Hence, the effect of η mainly1consists in modifying the region of existence of the type-I LSs, and eventually may imply their disappearance.While decreasing η to zero, high-order dispersionterms may become relevant, and should normally be in-cluded in the study. The next term to be considered cor-responds to the third-order dispersion effect. This termbreaks the reflection symmetry x → − x , inducing thedrift of the LSs and the modification of the collapsedsnaking as reported in [70]. Although these effects arevery relevant regarding real physical systems, their studyis beyond the scope of the present work, and will be ex-amined elsewhere. VI. BIFURCATION STRUCTURE OF TYPE-IILOCALIZED STATES
In this section we focus on the study of type-II LSs,and its bifurcation structure. As discussed previously,these states are formed through the locking of DWs ofdifferent polarities connecting − A + with A + . In contrastto the type-I states that exist in a reduced region aroundthe Maxwell point, type-II LSs live in a broader areain parameter space including regions II and III b . Whenapproaching ρ M in region II the type-II states becomea hybrid state formed by two type-I LSs related by thesymmetry A → − A . In what follows we will show howthis hybrid state also undergoes collapsed snaking. Inthis work we only consider stationary type-II LSs whichare formed through the locking of DWs of Ising type[58]. For high values of ρ , the DWs may undergo non-equilibrium Ising-Bloch transitions [58], resulting in thedrifting of LSs, domain oscillations, and complex dynam-ics that were studied in detail in [59].To start we fix (∆ , η ) = ( − , − . A → − A and separated by half of the domain size L/ ρ a , solutions of theform A ( x ) − A ( x + L/
2) exist, where A ( x ) is the weaklynonlinear solution about ρ a (44). This mixed state corre-sponds to the profile (vi) plotted on the red curve shownin Fig. 8(a) [see close-up view]. When moving upwardsalong the curve, each component of this mixed state be-haves as a single isolated state, undergoing collapsedsnaking. At each fold on the right a new dip is nucle-ated from the center of each structure resulting in thewidening of both states. This process can be seen in theprofiles (ix)-(xi).At this stage we can clearly identify the four DWsinvolved in the formation of the two LSs [see profile(xi)], which connect the CW solutions in the following sequence: − A + → A → A + → A → − A + . Increas-ing || A || further, the trivial state A decreases in width[see profiles (xii)-(xiv)] and eventually disappears. Thisoccurs approximately at the moment that A becomesunstable. As a result the two DWs − A + → A → A + be-come a single DW connecting − A + with A + , such that apair of type-I LSs transforms into the single type-II state(see panel (xv)). This type-II state persists for highervalues of ρ and extends to region II. The linear stabil-ity of these structure is shown in the close-up view ofFig. 8(a).We have verified that LSs of type-II with different ini-tial widths undergo a similar type of bifurcation struc-ture. To illustrate this behavior let us consider a sin-gle bump state, initially in region II, as the one plottedin panel (xvi). When modifying both ρ and || A || thisstructure is described by the bifurcation diagram plottedin purple in Fig. 8(b), where we also plotted the bifur-cation structure corresponding to the states (vi)-(xv) forcomparison (red diagram).When decreasing ρ , the LS (xvi) enters region III b where A is stable. Soon after that a plateau is createdaround A [see (xvii)] whose extension increases when ap-proaching ρ M [see (xviii)]. At this stage one can clearlyidentify two DWs connecting − A + with A and vice-versa, and the single bump type-II LSs becomes a pair oftype-I LSs consisting in a bright bump sitting on A atthe central position, and a dark wide LSs centered at dis-tance L/ ri [see profiles(xviii)-(xx)], until it becomes just a dark single bump.This hybrid state finally collides with the red bifurcationdiagram at SN r in symmetry-breaking pitchfork bifurca-tion. Indeed, at every SN ri other pitchfork bifurcationsoccurs from where branches mixed states solutions em-anates and undergo similar bifurcation structure, untilbecoming a type-II LS.We have confirmed that for higher values of | η | thebifurcation structure becomes much more complex, andtherefore the numerical continuation of the LSs is morecumbersome. Despite of this complexity, the connectionbetween type-I and II LSs persists and is qualitativelyequivalent to the one shown in Fig. 8. VII. LOCALIZED STRUCTURES IN THE (∆ , ρ ) − PARAMETER SPACE
In previous sections we have fixed ∆ = − η . However, inexperiments, η is normally fixed when choosing the fre-quency of the input pump field, and ∆ becomes one ofthe most relevant control parameter of the system. Be-cause of that here we study the effect that the modifi-cation of ∆ causes in the previously presented scenariowhen η is fixed to η = − .
25 0 25101 U , V (i)
25 0 25101 (ii)
25 0 25101 (iii)
25 0 25101 (iv)
25 0 25 τ (v)
25 0 25 τ U , V (vi)
25 0 25 x (vii)
25 0 25 x (viii)
25 0 25 x (ix)
25 0 25 x (x)
25 0 25 x U , V (xi)
25 0 25 x (xii)
25 0 25 x (xiii)
25 0 25 x (xiv)
25 0 25 x (xv)
25 0 25 x U , V (xvi)
25 0 25 x (xvii)
25 0 25 x (xviii)
25 0 25 x (xix)
25 0 25 x (xx) FIG. 8: (Color online) Bifurcation diagram for type-II LSs at (∆ , η ) = ( − , − . , ρ ) − pa-rameter space. Here, together with the pitchfork ρ a andsaddle-node ρ t bifurcation lines, we have added the linescorresponding to SN l and SN r (blue curves), the Maxwellpoint ρ M , and the MI ρ c corresponding to the chosenvalue η = − .
4. The gray area in-between SN l and SN r corresponds to the region where type-I DWs can lock andform LSs.When decreasing the absolute value of the ∆ , SN l andSN r approach one another and the gray region shrinksuntil it eventually disappears. SN l and SN r collide atthe Maxwell point and disappear in a cusp bifurcation3 − ∆ ρ SN l SN r ρ M ρ c ρ t ρ a III III a III b (i) (ii) (iii)(iv)
20 0 2001 U , V (i)
20 0 2001 (ii)
20 0 20 x U , V (iii)
20 0 20 x (iv) FIG. 9: (Color online) Phase diagram in the(∆ , ρ ) − parameter space for η = − .
4. The gray areabetween SN l and SN r corresponds to the region whereLSs of type-I exist. The pitchfork ρ a and saddle-node ρ t bifurcations of the CW solutions are plotted in blackand green solid lines respectively. The MI is the purpleline labeled by ρ c , and the Maxwell point of the system ρ M is the red solid line. The inset shows a close-up ofthe phase diagram around the cusp bifurcation C. Thelabels (i)-(ii), and (iii)-(iv) correspond to the LSs shownin the panels below for ∆ = −
2, and ∆ = − = 1 / √ t collides with A at ρ a .In contrast, increasing ∆ the region of existencewidens, and as a result it is easier to find LSs. We findthat LSs undergo the same type of collapsed snaking bi-furcation diagram while modifying ∆ , what shows thatthese type of solutions and their bifurcation structureare robust. Panels (i)-(iv) show the LSs correspondingto two fixed values of ∆ : profiles (i)-(ii) correspond to∆ = −
2, and (iii)-(iv) to ∆ = − , the mean field model (1) re-duces to a single PFGL equation with pure Kerr non-linearity that support analytical sech solutions of highamplitude [21]. Those solutions would correspond in ourwork to the single bump type-I LS shown in panel (iii)for a large enough ∆ . However, no analytical solutionhas been found for the wider LS (iv). Type-II LSs exist in region II and region III b for valuesof ρ above ρ M . However, for high values of | ∆ | their bi-furcation diagram can eventually become more complex. VIII. DISCUSSION
In this article we have presented a detailed and com-prehensive analysis of the bifurcation structure and sta-bility of LSs formed through locking of domain walls in χ (2) − dispersive cavities in the absence of walk-off. Todo so we have considered a degenerate optical paramet-ric oscillator in a doubly resonant configuration, and wehave focused on the sub-critical regime.To perform this analysis we have derived a PFGL typeof equation with a nonlocal nonlinearity [see Eq. (2)],which we have verified to reproduce the same results asthe full mean-field model (1) (Sec. II). In the PFGL con-text the pump field B is dynamically slaved to A [seeEq. (7)], and therefore it is characterized by the latter.In regions II and III b , the system is bistable and twotypes of DWs exist, forming connections between differ-ent CW solutions: i) A → A + , and ii) − A + → A + .We referred to these DWs as type-I and type-II. In thepresence of oscillatory tails, two DWs with different po-larities can lock forming LSs of different widths. We referto these LSs as type-I and II, depending on the type ofDW involved in their formation. We have shown thatLSs of type-I undergo collapsed snaking [62–64]. Here”collapsed” refers to the fact that the region of existenceof LSs shrinks exponentially as the width of the LS in-creases. Wider structures can only be found around theMaxwell point, and the observation of LSs with a singlebump is favored. Two examples of such type-I LSs areplotted in Fig. 10 for (∆ , η ) = ( − , −
4) using the vari-able A and B : Panel (i) shows the real and imaginarypart of A for a single bump LS, and in panel (ii) theslaved field B is plotted using the relation (7). Panels(iii) and (iv) represent the pump and signal fields corre-sponding to a wide structure.The collapsed snaking emerges from the pitchfork bi-furcation on A , and it connects back to the non-trivialCW state A + , either at the saddle-node SN t , or at the MI,depending on the control parameters of the system. Ap-plying multi-scale perturbation methods, we have beenable to calculate an analytical sech pulse-like solutionclose to the pitchfork bifurcation at ρ a .We have studied how the LSs and their associatedbifurcation structure are modified when the group ve-locity dispersion η changes. For doing so we fixed∆ = − η , ρ ) − parameter space shown in Fig. 7. The phase di-agram shows that when increasing | η | the type-I LSsdisappear, while type-II LSs persist in region II and III b well above the Maxwell point. In contrast, decreasing | η | , the region of existence of type-I LSs increases, andmany more type-I LSs can be found.When η →
0, the nonlocal nonlinear model (2) re-4
20 0 2050510 R e [ A ] , I m [ A ] (i)
20 0 2050510 R e [ B ] , I m [ B ] (ii)
20 0 20 x R e [ A ] , I m [ A ] (iii)
20 0 20 x R e [ B ] , I m [ B ] (iv) FIG. 10: (Color online) Real and imaginary parts of of A and B for two different types of LSs of type-I. Panels(i)-(iii) show the signal field A , and panels (ii)-(iv) thecorrespondent pump field B . Here (∆ , η ) = ( − , − | η | becomes very small, high-order dispersion effects mayplay an essential role and should be taken into account.In addition to the type-I LSs, a large variety of type-IILSs also formed through locking of DWs connecting theequivalent states − A + with A + . These states exist for awider range of parameters in region II and III b , and mayundergo non-equilibrium Ising-Bloch transitions [58], re-sulting in complex dynamics [59]. We have shown thatin region III b , every type-II LS becomes a hybrid statecomposed by two type-I LSs related by the symmetry A → − A and are separated by L/
2. Moreover, each ofthese states independently undergoes collapsed snakingaround the Maxwell point, which is also the bifurcationstructure characterizing its components.Finally, in Sec. VII, we have shown that type-I and IILSs persist for different values of ∆ , and that they aredescribed by the same kind of bifurcation structure. IX. CONCLUSION
The analysis presented in this paper provides a detailedstudy of the bifurcation structure and stability of the LSsarising in doubly resonant optical parametric oscillatorsin the absence of temporal walk-off. A potential physicalrealizable configuration for which the walk-off vanishes isdescribed in [35].The type of states studied here arise through the lock-ing of DWs formed between two continuous wave statesthat coexist in the same parameter range, i.e. in thepresence of bistability. The oscillatory damped nature of the DW interaction determines a particular bifurcationstructure known as collapsed snaking , which is genericand appears in a large number of systems in different con-texts [42, 62–64]. In contrast to the type-II LSs, whichhave been analyzed in detail in quadratic cavities [22, 23],as far as we known, the type-I LSs presented here havenot been reported elsewhere.To perform this analysis we have derived a nonlinearnonlocal model (2) similar to those derived for quadraticnonlinear cavities [31, 32, 34]. The results found herecan be extended to singly resonant cavities, where themodel is formally equivalent to (2), albeit with a differentnonlocal response [34].A natural extension of this work must include the ef-fect of the temporal walk-off, which breaks the x → − x symmetry inducing asymmetry and drift. We expect thatfor weak walk-off the collapsed snaking is modified in thesame fashion as in the context of Kerr cavities in thepresence of third-order dispersion [70].Quadratic dispersive cavities have gained a lot interestin the past few years as an alternative to Kerr cavities forthe generation of optical frequency combs [31–35]. There-fore, these results present a series of wave-forms whosefrequency spectrum could be of interest for applications. ACKNOWLEDGMENTS
We acknowledge the support from internal Funds fromKU Leuven and the FNRS (PPR), and funding fromthe European Research Council (ERC) under the Euro-pean Union’s Horizon 2020 research and innovation pro-gramme [Grant agreement No. 757800], (FL).
APPENDIX A: DERIVATION OF THEPARAMETRICALLY FORCEDGINZBURG-LANDAU EQUATION WITHNONLINEAR NONLOCAL COUPLING
In this Appendix we derive the Eq. (2) from the mean-field model (1). To do so we apply the same procedurethan in Refs. [32, 41]. This approach assumes the adia-batic elimination of the pump field B in Eq. (1b), i.e. B varies slowly with t , at least at time scale slower than the A field. Hence, one can assume that ∂ t B ≈ − (cid:0) α + i ∆ + d∂ x + iη ∂ x (cid:1) B + iA + S = 0 . (46)Defining the direct and inverse Fourier transforms F [ f ( x )]( k ) = (cid:90) ∞−∞ e ikx f ( x ) dx = ˜ f ( k ) , (47)and F − [ ˜ f ( k )]( x ) = 12 π (cid:90) ∞−∞ e − ikx ˜ f ( k ) dk = f ( x ) , (48)5one gets from Eq. (46) F [ B ] = i F [ J ] F [ A ] + F [ J ] F [ S ] (49)with F [ J ( k )] = 1 α + i (∆ + kd − η k ) . (50)Applying the inverse Fourier transform, Eq. (49) thenbecomes B ( x ) = i F − (cid:0) F ( J ) · F ( A ) (cid:1) + F − ( F ( J ) · F ( S )) . (51)Due to the convolution theorem, the first term on theright-hand side (rhs) of Eq. (51) becomes F − (cid:2) F ( J ) · F ( A ) (cid:3) = (cid:90) ∞−∞ J ( x (cid:48) ) A ( x − x (cid:48) ) dx (cid:48) = J ⊗ A , (52)where J ( x ) is the kernel defining a long-range nonlocalcoupling in x , and ⊗ stands for the convolution operation.With the definition of Dirac distribution δ ( k ) = 12 π (cid:90) ∞−∞ e ikx dx, (53)and taking ˜∆ = ∆ /α , the second term on (rhs) ofEq. (51) yields F − [ F ( J ) · F ( S )] = 2 πS F − [ F ( J ) · δ ( k )] = S F [ J (0)] = Sα (cid:113) e i atan( − ˜∆ ) (54)Thus the pump field finally reads, B ( x ) = iJ ⊗ A + ρe i atan( − ˜∆ ) , (55)where we have defined ρ = Sα (cid:113) . (56)Inserting (55) into Eq. (1a), the later becomes in PFGLtype of equation with a nonlinear nonlocal long rangeinteraction term: ∂ t A = − (1+ i ∆ ) A − iη ∂ x A − ¯ A ( J ⊗ A )+ ρ ¯ Ae iψ , (57)with ψ = π/ − ˜∆ ) / . (58)Rescaling the A field as A = A e iψ (cid:113) α (1 + ˜∆ ) (59)the Eq. (57) then becomes ∂ t A = − (1 + i ∆ ) A − iη ∂ x A − ¯ A ( A ⊗ J ) + ρ ¯ A . (60) With this normalization the long-range interaction kernelbecomes J ( x ) = 1 + ˜∆ π (cid:90) ∞−∞ e − ikx dx i ( ˜∆ + γk − ˜ η k ) , (61)with γ = d/α , and ˜ η = η /α . The real and imaginaryparts of this kernel are J R ( x ) = 1 + ˜∆ π (cid:90) ∞−∞ e − ikx dk + γk − ˜ η k ) , (62) J I ( x ) = − π (cid:90) ∞−∞ ( ˜∆ + γk − ˜ η k ) e − ikx dx + γk − ˜ η k ) . (63)With this normalization the B field becomes B = ( − A ⊗ J + ρ ) e i atan( − ˜∆ ) (64)In this work we consider γ = 0, and therefore (62) and(63) are symmetric under the transformation x → − x .The square root factor in Eq. (59) has been introducedfor convenience in order to obtain the standard form ofthe PFGL Eq. (45) in the limit γ, ˜ η → APPENDIX B: WEAKLY NONLINEARANALYSIS AROUND THE PITCHFORKBIFURCATION
In this Appendix we show how to obtain the station-ary amplitude equation (38) around ρ a starting fromthe equation at order (cid:15) in the perturbation expansion,namely: L (cid:20) u v (cid:21) = − ( L + N ) (cid:20) u v (cid:21) . (65)To solve this equation we have first to deal with the non-linear nonlocal operator N = (cid:15) N = − (cid:20) N a N b N b −N a (cid:21) , (66)with N a = u ⊗ J R − v ⊗ J R − u v ⊗ J I , (67a) N b = u ⊗ J I − v ⊗ J I + 2 u v ⊗ J R , (67b)where the solution of the problem at O ( (cid:15) ) reads (cid:20) u v (cid:21) = (cid:20) ξ (cid:21) a ( x ) , (68)with ξ = ∆ / (1 − ρ a ) and a ( x ) a real function.At this point we have to evaluate the convolutionterms, and for doing so we follow the procedure describedin Ref. [60, 71]. In order to perform this calculation we6consider that all the terms posed on the long length scale x are assumed to be almost constant over the regionwhere the kernel J is large, what is equivalent to con-sider for a very narrow kernel [45]. This makes sensewhen one assumes that the amplitude a of the envelopeis smooth, and the kernel decays much more rapidly thanthe envelope.With these considerations we obtain: u ⊗ J R = (cid:90) ∞−∞ u ( x (cid:48) ) J R ( x − x (cid:48) ) dx (cid:48) = ξ (cid:90) ∞−∞ a ( x (cid:48) ) J R ( x − x (cid:48) ) dx (cid:48) ≈ ξ a ( x ) (cid:90) ∞−∞ J R ( x − x (cid:48) ) dx (cid:48) = ξ a ( x ) F − (2 πδ ( k ) F [ J R ]( k )) = ξ a ( x ) F [ J R ](0) = ξ a ( x ) ,u v ⊗ J I = (cid:90) ∞−∞ u ( x (cid:48) ) v ( x (cid:48) ) J I ( x − x (cid:48) ) dx (cid:48) ≈ ξa ( x ) (cid:90) ∞∞ J I ( x − x (cid:48) ) dx (cid:48) = ξa ( x ) F [ J I ](0) = − ξ ∆ a ( x ) , and with the same approach v ⊗ J R ≈ a ( x ) F [ J R ](0) = a ( x ) , u ⊗ J I ≈ − ∆ ξ a ( x ) ,v ⊗ J I ≈ − ∆ a ( x ) ,u v ⊗ J R ≈ ξa ( x ) , Thus, the the components of the nonlinear operator (66)become N a = ( ξ + 2 ξ ∆ − a ( x ) (71a) N b = ( − ξ ∆ + 2 ξ + ∆ ) a ( x ) (71b)The amplitude equation about ρ a is then obtained fromthe solvability condition w T · L (cid:20) u v (cid:21) + w T · N (cid:20) u v (cid:21) = (cid:20) (cid:21) , (72)where w T = [ − ξ, L † w = 0.The evaluation of the first term yields w T · L (cid:20) u v (cid:21) = − δ ( ξ + 1) a − ξη ∂ x a, (73)while the second one gives w T · N (cid:20) u v (cid:21) = ( ξ + 1)( ξ + 2∆ ξ − a ( x ) . (74)After arranging these terms and simplifying them onegets the stationary amplitude equation (38) about ρ a . [1] P. Coullet, C. Elphick, and D. Repaux, “Nature of spatialchaos,” Physical Review Letters , vol. 58, pp. 431–434,Feb. 1987.[2] P. Coullet, “Localized patterns and fronts in nonequilib-rium systems,”
International Journal of Bifurcation andChaos , vol. 12, pp. 2445–2457, Nov. 2002.[3] M. Tlidi, P. Mandel, and R. Lefever, “Localized struc-tures and localized patterns in optical bistability,”
Phys-ical Review Letters , vol. 73, pp. 640–643, Aug. 1994.[4] Fernandez-Oto C., Tlidi M., Escaff D., and Clerc M. G.,“Strong interaction between plants induces circular bar-ren patches: fairy circles,”
Philosophical Transactions ofthe Royal Society A: Mathematical, Physical and Engi-neering Sciences , vol. 372, p. 20140009, Oct. 2014.[5] D. Ruiz-Reyn´es, D. Gomila, T. Sintes, E. Hern´andez-Garc´ıa, N. Marb`a, and C. M. Duarte, “Fairy circlelandscapes under the sea,”
Science Advances , vol. 3,p. e1603262, Aug. 2017.[6] H. Willebrand, M. Or-Guil, M. Schilke, H. G. Purwins,and Y. A. Astrov, “Experimental and numerical observa-tion of quasiparticle like structures in a distributed dissi- pative system,”
Physics Letters A , vol. 177, pp. 220–224,June 1993.[7] P. B. Umbanhowar, F. Melo, and H. L. Swinney, “Local-ized excitations in a vertically vibrated granular layer,”
Nature , vol. 382, p. 793, Aug. 1996.[8] V. B. Taranenko, I. Ganne, R. J. Kuszelewicz, and C. O.Weiss, “Patterns and localized structures in bistablesemiconductor resonators,”
Physical Review A , vol. 61,p. 063818, May 2000.[9] P. L. Ramazza, S. Ducci, S. Boccaletti, and F. T. Arec-chi, “Localized versus delocalized patterns in a nonlinearoptical interferometer,”
Journal of Optics B: Quantumand Semiclassical Optics , vol. 2, pp. 399–405, June 2000.[10] S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato,S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tis-soni, T. Kn¨odl, M. Miller, and R. J¨ager, “Cavity soli-tons as pixels in semiconductor microcavities,”
Nature ,vol. 419, p. 699, Oct. 2002.[11] F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit,and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nature Photonics , vol. 4, pp. 471–476, July 2010.[12] G. Nicolis and I. Prigogine,
Self-organization in nonequi-librium systems: from dissipative structures to orderthrough fluctuations . New York, N.Y.: Wiley, 1977.OCLC: 797228045.[13] N. Akhmediev and A. Ankiewicz, eds.,
Dissipative Soli-tons . Lecture Notes in Physics, Berlin Heidelberg:Springer-Verlag, 2005.[14] J. D. Murray,
Mathematical Biology: I. An Introduc-tion . Interdisciplinary Applied Mathematics, Mathemat-ical Biology, New York: Springer-Verlag, 3 ed., 2002.[15] M. G. Clerc, D. Escaff, and V. M. Kenkre, “Patternsand localized structures in population dynamics,”
Phys-ical Review E , vol. 72, p. 056217, Nov. 2005.[16] A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi,R. Lefever, and L. A. Lugiato, “Pattern formation in apassive Kerr cavity,”
Chaos, Solitons & Fractals , vol. 4,pp. 1323–1354, Aug. 1994.[17] W. J. Firth and A. Lord, “Two-dimensional solitonsin a Kerr cavity,”
Journal of Modern Optics , vol. 43,pp. 1071–1077, May 1996.[18] C. Etrich, U. Peschel, and F. Lederer, “Solitary Wavesin Quadratically Nonlinear Resonators,”
Physical ReviewLetters , vol. 79, pp. 2454–2457, Sept. 1997.[19] K. Staliunas and V. J. S´anchez-Morcillo, “Localizedstructures in degenerate optical parametric oscillators,”
Optics Communications , vol. 139, pp. 306–312, July1997.[20] K. Staliunas and V. J. S´anchez-Morcillo, “Spatial-localized structures in degenerate optical parametric os-cillators,”
Physical Review A , vol. 57, pp. 1454–1457, Feb.1998.[21] S. Longhi, “Localized structures in optical parametricoscillation,”
Physica Scripta , vol. 56, pp. 611–618, Dec.1997.[22] G.-L. Oppo, A. J. Scroggie, and W. J. Firth, “From do-main walls to localized structures in degenerate opticalparametric oscillators,”
Journal of Optics B: Quantumand Semiclassical Optics , vol. 1, pp. 133–138, Jan. 1999.[23] G.-L. Oppo, A. J. Scroggie, and W. J. Firth, “Character-ization, dynamics and stabilization of diffractive domainwalls and dark ring cavity solitons in parametric oscilla-tors,”
Physical Review E , vol. 63, May 2001.[24] K. Staliunas and V. J. S´anchez-Morcillo,
Transverse Pat-terns in Nonlinear Optical Resonators . Springer Tractsin Modern Physics, Berlin Heidelberg: Springer-Verlag,2003.[25] Y. K. Chembo and C. R. Menyuk, “SpatiotemporalLugiato-Lefever formalism for Kerr-comb generation inwhispering-gallery-mode resonators,”
Physical Review A ,vol. 87, p. 053852, May 2013.[26] F. Leo, L. Gelens, P. Emplit, M. Haelterman, andS. Coen, “Dynamics of one-dimensional Kerr cavity soli-tons,”
Optics Express , vol. 21, pp. 9180–9191, Apr. 2013.[27] T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kon-dratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Tem-poral solitons in optical microresonators,”
Nature Pho-tonics , vol. 8, pp. 145–152, Feb. 2014.[28] P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken,R. Holzwarth, and T. J. Kippenberg, “Optical frequencycomb generation from a monolithic microresonator,”
Na-ture , vol. 450, pp. 1214–1217, Dec. 2007.[29] T. J. Kippenberg, R. Holzwarth, and S. A. Diddams,“Microresonator-Based Optical Frequency Combs,”
Sci- ence , vol. 332, pp. 555–559, Apr. 2011.[30] A. Pasquazi, M. Peccianti, L. Razzari, D. J. Moss,S. Coen, M. Erkintalo, Y. K. Chembo, T. Hansson,S. Wabnitz, P. Del’Haye, X. Xue, A. M. Weiner, andR. Morandotti, “Micro-combs: A novel generation of op-tical sources,”
Physics Reports , vol. 729, pp. 1–81, Jan.2018.[31] F. Leo, T. Hansson, I. Ricciardi, M. De Rosa, S. Coen,S. Wabnitz, and M. Erkintalo, “Walk-Off-Induced Modu-lation Instability, Temporal Pattern Formation, and Fre-quency Comb Generation in Cavity-Enhanced Second-Harmonic Generation,”
Physical Review Letters , vol. 116,p. 033901, Jan. 2016.[32] F. Leo, T. Hansson, I. Ricciardi, M. De Rosa, S. Coen,S. Wabnitz, and M. Erkintalo, “Frequency-comb forma-tion in doubly resonant second-harmonic generation,”
Physical Review A , vol. 93, p. 043831, Apr. 2016.[33] S. Mosca, M. Parisi, I. Ricciardi, F. Leo, T. Hansson,M. Erkintalo, P. Maddaloni, P. D. Natale, S. Wabnitz,S. Wabnitz, and M. D. Rosa, “Frequency comb genera-tion in continuously pumped optical parametric oscilla-tor,” in
Frontiers in Optics 2017 (2017), paper FTh2B.4 ,p. FTh2B.4, Optical Society of America, Sept. 2017.[34] S. Mosca, M. Parisi, I. Ricciardi, F. Leo, T. Hansson,M. Erkintalo, P. Maddaloni, P. De Natale, S. Wabnitz,and M. De Rosa, “Modulation Instability Induced Fre-quency Comb Generation in a Continuously PumpedOptical Parametric Oscillator,”
Physical Review Letters ,vol. 121, p. 093903, Aug. 2018.[35] T. Hansson, P. Parra-Rivas, M. Bernard, F. Leo, L. Ge-lens, and S. Wabnitz, “Quadratic soliton combs in dou-bly resonant second-harmonic generation,”
Optics Let-ters , vol. 43, pp. 6033–6036, Dec. 2018.[36] S. Trillo, M. Haelterman, and A. Sheppard, “Stable topo-logical spatial solitons in optical parametric oscillators,”
Optics Letters , vol. 22, pp. 970–972, July 1997.[37] P. Parra-Rivas, L. Gelens, T. Hansson, S. Wabnitz, andF. Leo, “Frequency comb generation through the lock-ing of domain walls in doubly resonant dispersive opticalparametric oscillators,”
Optics Letters , vol. 44, pp. 2004–2007, Apr. 2019.[38] M. Haelterman, S. Trillo, and S. Wabnitz, “Dissipativemodulation instability in a nonlinear dispersive ring cav-ity,”
Optics Communications , vol. 91, pp. 401–407, Aug.1992.[39] G.-L. Oppo, M. Brambilla, and L. A. Lugiato, “Forma-tion and evolution of roll patterns in optical parametricoscillators,”
Physical Review A , vol. 49, pp. 2028–2032,Mar. 1994.[40] R. Zambrini, M. San Miguel, C. Durniak, and M. Taki,“Convection-induced nonlinear symmetry breaking inwave mixing,”
Physical Review E , vol. 72, p. 025603, Aug.2005.[41] N. I. Nikolov, D. Neshev, O. Bang, and W. Z.Kr´olikowski, “Quadratic solitons as nonlocal solitons,”
Physical Review E , vol. 68, Sept. 2003.[42] J. Burke, A. Yochelis, and E. Knobloch, “Classificationof Spatially Localized Oscillations in Periodically ForcedDissipative Systems,”
SIAM Journal on Applied Dynam-ical Systems , vol. 7, pp. 651–711, Jan. 2008.[43] Q. Lin and G. P. Agrawal, “Raman response function forsilica fibers,”
Optics Letters , vol. 31, pp. 3086–3088, Nov.2006.[44] Y. K. Chembo, I. S. Grudinin, and N. Yu, “Spatiotempo- ral dynamics of Kerr-Raman optical frequency combs,” Physical Review A , vol. 92, p. 043818, Oct. 2015.[45] W. Z. Krolikowski, O. Bang, W. Krolikowski, andJ. Wyller, “Modulational instability in nonlocal nonlin-ear Kerr media.,”
Physical review. E, Statistical, nonlin-ear, and soft matter physics , vol. 64, no. 1, p. 016612,2001.[46] D. Suter and T. Blasberg, “Stabilization of transversesolitary waves by a nonlocal response of the nonlinearmedium,”
Physical Review A , vol. 48, pp. 4583–4587,Dec. 1993.[47] W. J. Firth, L. Columbo, and A. J. Scroggie, “Pro-posed Resolution of Theory-Experiment Discrepancy inHomoclinic Snaking,”
Physical Review Letters , vol. 99,p. 104503, Sept. 2007.[48] J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S.Porto, and J. R. Whinnery, “Long-Transient Effects inLasers with Inserted Liquid Samples,”
Journal of AppliedPhysics , vol. 36, pp. 3–8, Jan. 1965.[49] L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, andR. J. Horowicz, “Bistability, self-pulsing and chaos in op-tical parametric oscillators,”
Il Nuovo Cimento D , vol. 10,pp. 959–977, Aug. 1988.[50] G. J. de Valc´arcel, K. Staliunas, E. Rold´an, and V. J.S´anchez-Morcillo, “Transverse patterns in degenerate op-tical parametric oscillation and degenerate four-wavemixing,”
Physical Review A , vol. 54, pp. 1609–1624, Aug.1996.[51] M. Haragus and G. Iooss,
Local Bifurcations, CenterManifolds, and Normal Forms in Infinite-DimensionalDynamical Systems . Universitext, London: Springer-Verlag, 2011.[52] Turing Alan Mathison, “The chemical basis of morpho-genesis,”
Philosophical Transactions of the Royal Societyof London. Series B, Biological Sciences , vol. 237, pp. 37–72, Aug. 1952.[53] M. Tlidi, P. Mandel, and M. Haelterman, “Spatiotempo-ral patterns and localized structures in nonlinear optics,”
Physical Review E , vol. 56, pp. 6524–6530, Dec. 1997.[54] M. Tlidi and M. Haelterman, “Robust Hopf-Turingmixed-mode in optical frequency conversion systems,”
Physics Letters A , vol. 239, pp. 59–64, Feb. 1998.[55] A. R. Champneys, “Homoclinic orbits in reversible sys-tems and their applications in mechanics, fluids andoptics,”
Physica D: Nonlinear Phenomena , vol. 112,pp. 158–186, Jan. 1998.[56] J. M. Chomaz, “Absolute and convective instabilitiesin nonlinear systems,”
Physical Review Letters , vol. 69,pp. 1931–1934, Sept. 1992.[57] S. M. Allen and J. W. Cahn, “A microscopic theory forantiphase boundary motion and its application to an-tiphase domain coarsening,”
Acta Metallurgica , vol. 27, pp. 1085–1095, June 1979.[58] P. Coullet, J. Lega, B. Houchmandzadeh, and J. Lajze-rowicz, “Breaking chirality in nonequilibrium systems,”
Physical Review Letters , vol. 65, pp. 1352–1355, Sept.1990.[59] D. Gomila, P. Colet, and D. Walgraef, “Theory for theSpatiotemporal Dynamics of Domain Walls close to aNonequilibrium Ising-Bloch Transition,”
Physical ReviewLetters , vol. 114, p. 084101, Feb. 2015.[60] D. Morgan and J. H. P. Dawes, “The Swift–Hohenbergequation with a nonlocal nonlinearity,”
Physica D: Non-linear Phenomena , vol. 270, pp. 60–80, Mar. 2014.[61] E. L. Allgower and K. Georg,
Numerical ContinuationMethods: An Introduction . Springer Series in Computa-tional Mathematics, Berlin Heidelberg: Springer-Verlag,1990.[62] J. Knobloch and T. Wagenknecht, “Homoclinic snakingnear a heteroclinic cycle in reversible systems,”
PhysicaD: Nonlinear Phenomena , vol. 206, pp. 82–93, June 2005.[63] Y. P. Ma, J. Burke, and E. Knobloch, “Defect-mediatedsnaking: A new growth mechanism for localized struc-tures,”
Physica D: Nonlinear Phenomena , vol. 239,pp. 1867–1883, Oct. 2010.[64] P. Parra-Rivas, E. Knobloch, D. Gomila, and L. Gelens,“Dark solitons in the Lugiato-Lefever equation with nor-mal dispersion,”
Physical Review A , vol. 93, p. 063839,June 2016.[65] C. Godey, “A bifurcation analysis for the Lugiato-Lefeverequation,”
The European Physical Journal D , vol. 71,p. 131, May 2017.[66] P. Parra-Rivas, D. Gomila, L. Gelens, and E. Knobloch,“Bifurcation structure of localized states in the Lugiato-Lefever equation with anomalous dispersion,”
PhysicalReview E , vol. 97, p. 042204, Apr. 2018.[67] G. Kozyreff and S. J. Chapman, “Asymptotics of LargeBound States of Localized Structures,”
Physical ReviewLetters , vol. 97, p. 044502, July 2006.[68] P. D. Woods and A. R. Champneys, “Heteroclinic tan-gles and homoclinic snaking in the unfolding of a degen-erate reversible Hamiltonian–Hopf bifurcation,”
PhysicaD: Nonlinear Phenomena , vol. 129, pp. 147–170, May1999.[69] J. Burke and E. Knobloch, “Snakes and ladders: Local-ized states in the Swift–Hohenberg equation,”
PhysicsLetters A , vol. 360, pp. 681–688, Jan. 2007.[70] P. Parra-Rivas, D. Gomila, and L. Gelens, “Coexis-tence of stable dark- and bright-soliton Kerr combsin normal-dispersion resonators,”
Physical Review A ,vol. 95, p. 053863, May 2017.[71] C. Kuehn and S. Throm, “Validity of amplitude equa-tions for nonlocal nonlinearities,”