aa r X i v : . [ c ond - m a t . qu a n t - g a s ] M a r Gap solitons in Rabi lattices
Zhaopin Chen and Boris A. Malomed , Department of Physical Electronics, School of Electrical Engineering,Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel Laboratory of Nonlinear-Optical Informatics, ITMO University, St. Petersburg 197101, Russia
We introduce a two-component one-dimensional system, which is based on two nonlinearSchr¨odinger/Gross-Pitaevskii equations (GPEs) with spatially periodic modulation of linear cou-pling (“Rabi lattice”) and self-repulsive nonlinearity. The system may be realized in a binaryBose-Einstein condensate, whose components are resonantly coupled by a standing optical wave, aswell as in terms of the bimodal light propagation in periodically twisted fibers. The system sup-ports various types of gap solitons (GSs), which are constructed, and their stability is investigated,in the first two finite bandgaps of the underlying spectrum. These include on- and off-site-centeredsolitons (the GSs of the off-site type are additionally categorized as spatially even and odd ones),which may be symmetric or antisymmetric, with respect to the coupled components. The GSs arechiefly stable in the first finite bandgap, and unstable in the second one. In addition to that, thereare narrow regions near the right edge of the first bandgap, and in the second one, which featureintricate alternation of stability and instability. Unstable solitons evolve into robust breathers orspatially confined turbulent modes. On-site-centered GSs are also considered in a version of thesystem which is made asymmetric by the Zeeman effect, or by birefringence of the optical fiber. Aregion of alternate stability is found in the latter case too. In the limit of strong asymmetry, GSsare obtained in a semi-analytical approximation, which reduces two coupled GPEs to a single onewith an effective lattice potential.
PACS numbers: 42.65.Tg; 42.70.Qs; 05.45.Yv
I. INTRODUCTION
Solitons in lattice potentials have drawn a great deal of interest in recent decades, as they occur in diverse physicalsettings, and exist in many different varieties [1]-[4]. Periodic potentials, added to the underlying nonlinear Schr¨odingeror Gross-Pitaevskii equations (GPEs), help to create and stabilize solitons which do not exist or are unstable in freespace. In optics, one source of the spatial periodicity is provided by Bragg gratings [5]. Effective periodic potentialscan also be readily induced by photonic crystals, which can be built as permanent structures by means of varioustechniques [6]-[8], or as virtual lattices formed by interfering laser beam in photorefractive crystals [6, 9]. For matterwaves in atomic Bose-Einstein condensates (BECs), perfect periodic potentials are imposed by optical lattices, i.e.,interference patterns constructed by counter-propagating coherent optical beams [1, 10]. In the presence of the self-repulsive nonlinearity, localized modes which self-trap in periodic potentials are usually called gap solitons (GSs),as they exist in bandgaps of the underlying Bloch spectrum induced by the potential in the linear approximation[11]-[15]. Different families of GSs are distinguished by the number of the bandgaps in which they reside.In the presence of an appropriate periodic potential, binary BECs [16] and binary photonic systems [17, 19] can hosttwo-component GSs, which have been theoretically elaborated in various settings [17]-[23]. The use of the Feshbach-resonance technique [26], that switches the repulsion between atoms into attraction, makes it possible to createtwo-component symbiotic solitons in binary BEC, which are supported by attraction between the two components,while each of them is subject to self-repulsion [25]. This concept was extended to symbiotic GSs in a system of twomutually repelling components loaded into a common lattice potential [23]. Another extension of this concept waselaborated for dark solitons in spinor systems [24].An essential ingredient of many two-component systems is linear interconversion (Rabi coupling) between thecomponents. In binary BEC, the interconversion is driven by a resonant electromagnetic field, which couples differentatomic states representing the components [27, 28]. Two-component GSs coupled by the linear interconversion werestudied too [22, 29]. In optics, the Rabi coupling is emulated by the linear coupling between copropagating wavesin dual-core waveguides [30]. In particular, GSs in a dual-core Bragg grating were studied in Ref. [17]. Similarly,a dual-core BEC trap may hold two matter-wave fields with an effective Rabi coupling between them, provided bytunneling of atoms across a gap separating the two cores [31]. In this connection, it is relevant to mention that GSsmay be supported by
Zeeman lattices , i.e., spatially periodic modulation of the difference in the chemical potentialbetween two BEC components which are linearly coupled by spin-orbit coupling [32].In the present work, our objective is to propose a different mechanism of the creation of two-component GSs,without the use of any lattice potential, but rather making use of the linear coupling between two components ofthe wave field in the form of a standing wave, which may be called a “Rabi lattice” (cf. “Rabi management”, i.e.,the linear coupling with a time-periodic coefficient, introduced in Ref. [28]), assuming intrinsic self-repulsion in eachcomponent. This setting may be realized in a binary BEC illuminated by a pair of counterpropagating resonantly-coupling waves, the interference of which builds the standing wave. In this connection, it is relevant to mention recentwork [18], in which it was demonstrated that a spatially localized (rather than periodically modulated) linear couplingbetween the components may play a role in the soliton dynamics similar to that of localized attractive potentials. Wehere focus on the basic case of the linear coupling, neglecting nonlinear cross-repulsion between the components (itcan be suppressed by means of the Feshbach resonance [33]). An extended version of the system includes asymmetrybetween the two components, which may be imposed by the Zeeman splitting between them. As shown in the nextsection, a similar model can be implemented in nonlinear optics, considering the co-propagation of two polarizationsin a periodically twisted fibers, while the asymmetry may be imposed by the fiber’s intrinsic birefringence.In the symmetric system, the equations for the two components merge into a single one, if solutions with equalcomponents are looked for. In this case, the standing-wave-shaped linear coupling turns into an effective latticepotential. Although shapes of the corresponding GSs are known, a new problem is their stability in the framework ofthe two-component system, as well as constructing GSs in the asymmetric one. In this work, we concentrate on the1D setting, while the 2D version, which is possible in BEC, will be considered elsewhere.The rest of the paper is structured as follows. In Sec. II, we introduce the model and present a method for the studyof stability of the GSs. Several families of GSs and results for their stability are produced in Sec. III. The solitonsare classified as on- and off-site-centered ones and symmetric or antisymmetric, with respect to the two component.The off-site-centered GSs are additionally categorized as spatially even or odd modes. In Sec. IV, we address on-site-centered GSs in an asymmetric version of the system, with the objective to identify their existence and stabilityareas. For a strongly asymmetric system, analytical approximation is developed too. The paper is concluded by Sec.V.
II. THE MODEL
The two-component system with the linear-coupling coefficient spatially shaped as the standing wave (Rabi lattice)is based on the system of scaled GPEs for two components of the mean-field wave function, u ( x, t ) and v ( x, t ): (cid:18) i ∂∂t + 12 ∂ ∂x − σ | u | (cid:19) u + ǫ cos(2 x ) · v + bu = 0 , (1) (cid:18) i ∂∂t + 12 ∂ ∂x − σ | v | (cid:19) v + ǫ cos(2 x ) · u − bv = 0 . (2)where ǫ is the amplitude of the Rabi lattice, whose period is fixed to be π by making use of obvious rescaling, andreal coefficient b accounts for possible asymmetry introduced by the Zeeman splitting. Results are reported below for ǫ = 6 in Eqs. (1) and (2), which adequately represents the generic situation.Further, σ = +1 and − σ = +1, which cannot create regular solitons in free space.Effects of the nonlinear interaction between the two components, with relative strength g (the use of the Feshbachresonance makes it relevant to consider all the cases of g > g = 0, and g < | u | u → (cid:0) | u | + g | v | (cid:1) u, | v | v → (cid:0) | v | + g | u | (cid:1) v. (3)We chiefly disregard the XPM terms here (as mentioned above, in BEC this interaction may be eliminated with thehelp of the Feshbach-resonance method), except for the limit case of strong asymmetry [large b in Eqs. (1) and (2)],in which the XPM can be easily taken into account in the framework of the semi-analytical approximation, see Eqs.(12)-(14) below. In a systematic form, XPM effects will be considered elsewhere.Unlike the BEC system, introduction of a similar model in terms of optical dual-core waveguides is problematic, asthe coefficient of the inter-core coupling cannot, normally, change its sign. On the other hand, the same linear couplingas defined in Eqs. (1) and (2) may naturally appear in the model of the co-propagation of two linear polarizations oflight in a “rocking” optical fiber, subject to a periodically modulated twist [34] (with evolution variable t replaced bythe propagation distance, z , and x replaced by the reduced temporal variable, τ ), while parameter b represents thephase-velocity birefringence of the fiber. In the latter case, however, the XPM terms with g = 2 / u τ , − v τ ) inthe respective equations, although this effect is usually much weaker than the phase-velocity birefringence. FIG. 1: (Color online) The bandgap spectrum of the linearized version of Eqs. (1) and (2) with ǫ = 6, for b = 0 (a) and b = 3(b), which correspond, respectively, to the symmetric and asymmetric system. Shaded areas designate Bloch bands where gapsolitons do not exist. The semi-infinite gap and lowest finite ones are labeled. Stationary solutions of Eqs. (1) and (2) are looked for as usual, u ( x, t ) = e − iµt U ( x ) , v ( x, t ) = e − iµt V ( x ) , (4)where µ is a real chemical potential, and real wave functions U and V obey the stationary equations, as said above: µU + 12 U ′′ − U + ǫ cos(2 x ) · V + bU = 0 , (5) µV + 12 V ′′ − V + ǫ cos(2 x ) · U − bV = 0 , (6)where the prime stands for d/dx (hereafter, we fix the defocusing sign of the nonlinearity, σ = +1, as said above).Numerical solutions of Eqs. (5) and (6) was produced by means of the Newton’s method. The bandgap spectrumgenerated by the solution of the linearized version of the equations is shown in Fig. 1.Equations (1) and (2) conserve the total norm, i.e., scaled number of atoms, in terms of BEC, N = Z + ∞−∞ (cid:0) | u | + | v | (cid:1) dx ≡ N u + N v , (7)and the Hamiltonian, H = Z + ∞−∞ " (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂x (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∂v∂x (cid:12)(cid:12)(cid:12)(cid:12) + | u | + | v | ! − ǫ ( u ∗ v + uv ∗ ) cos (2 x ) + b (cid:0) | v | − | u | (cid:1)(cid:3) dx . (8)where ∗ stands for the complex conjugate, while the conservation of the momentum is destroyed by the presence ofthe Rabi lattice. Asymmetry of two-component solitons is determined by the respective ratio, R = ( N u − N v ) / ( N u + N v ) . (9)For families of GSs, N and H , as well as R (if R = 0), may be naturally considered as functions of chemical potential µ . Stability of stationary solutions can be investigated by means of the linearization against small complex perturba-tions φ , ( x, z ) and ψ , ( x, z ) added to solution (4): u ( x, t ) = e − iµt [ U ( x ) + φ ( x ) e − iλt + φ ∗ ( x ) e iλ ∗ t ] ,v ( x, t ) = e − iµt [ V ( x ) + ψ ( x ) e − iλt + ψ ∗ ( x ) e iλ ∗ t ] , (10)where λ is the perturbation eigenfrequency, which may be complex. Instability takes place if there is at least oneeigenvalue with Im( λ ) >
0. Oscillatory instabilities correspond to complex λ , with both real and imaginary partsdifferent from zero, which happens in previously studied related systems [14, 23]. The substitution of expressions (10)into Eqs. (1), (2) and subsequent linearization leads to the eigenvalue problem for λ , based on the following systemof equations: − φ ′′ + U (2 φ + φ ) − ( b + µ ) φ − ǫ cos(2 x ) ψ = λφ , φ ′′ − U (2 φ + φ ) + ( b + µ ) φ + ǫ cos(2 x ) ψ = λφ , − ψ ′′ + V (2 ψ + ψ ) + ( b − µ ) ψ − ǫ cos(2 x ) φ = λψ , ψ ′′ − V (2 ψ + ψ ) − ( b − µ ) ψ + ǫ cos(2 x ) φ = λψ . (11)These equations can be rewritten in the matrix form, ˆ M ( φ , φ , ψ , ψ ) T = λ ( φ , φ , ψ , ψ ) T , where operator ˆ M corresponds to the matrix in the left-hand side of Eqs. (11). For the numerical solution of the stability problem,we discretize functional expressions in matrix elements by means of the center-difference numerical scheme, and thencalculate the eigenvalue spectrum of the matrix, truncated to a sufficiently large finite size, for stationary solutions. III. GAP SOLITONS
The numerical solution of the symmetric version of Eqs. (5) and (6), with b = 0, shows that the system producesseveral basic types of GSs. With respect to their spatial structure, they can be classified as on- and off-site-centeredlocalized modes, which feature, respectively, a single density maximum coinciding with a local minimum of themodulation function, − ǫ cos(2 x ), i.e., x = 0, or two density maxima placed at adjacent modulation maxima, x = ± π/
2. Further, the off-cite-centered modes, with the pair of density maxima, may be spatially even or odd asfunctions of x (all onsite-centered modes are, obviously, of the even type). Then, in the absence of the Zeemansplitting ( b = 0), the two-component GSs are categorized as “symmetric” or “antisymmetric”, if their two componentsare, respectively, identical, or differ by opposite signs. In the following analysis, we first focus on on-site-centeredsymmetric, off-site-centered even antisymmetric, and off-sited odd antisymmetric GS species. We also consider on-site-centered antisymmetric, off-site-centered even symmetric, and off-site-centered odd symmetric modes, whose stabilityis, severally, the same as that of the three above-mentioned species. The analysis is performed for the GSs residingin the first and second finite bandgaps, see Fig. 1. A. On-site-centered symmetric gap solitons
It is obvious that on-site-centered symmetric GSs [see typical examples in Figs. 2(a,b) and 3(a,b)], with equalcomponents, U ( x ) = V ( x ) (in the system with b = 0), are identical, in their shape, to the GSs produced by thesingle-component GPE, the Rabi lattice becoming equivalent to a single-component lattice potential, − ǫ cos (2 x ).Accordingly, the density maximum placed at x = 0 in Figs. 2(a) and 3(a) tends to minimize the effective potential.However, the difference in the stability between the single-component model and the present two-component system isessential, as shown in Fig. 4(a,b). Recall that the GS family tends to be stable in the first two finite bandgaps of thesingle-component model, except for a weak oscillatory instability, caused by the appearance of complex eigenvalues,close to the right edge of the first bandgap, and in the second one [36]. In addition, dashed magenta lines display, inboth panels (a) and (b) of Fig. 4, analytical predictions based on the Thomas-Fermi approximation (see details inthe figure caption), which was elaborated for single-component GSs in Ref. [23]Figures 4(a,b) demonstrate that the on-site-centered GSs are stable in the first finite bandgap of the two-componentsystem (see a typical example in Fig. 2), except for a small region near the right edge of the bandgap, which roughlyresembles the above-mentioned weak oscillatory instability of single-component GSs near the edge of the first bandgap[36]. In this small region, stable solitons alternate with unstable ones, which are subject to a weak oscillatory instabilitycaused by complex eigenvalues. A detailed structure of this region is displayed in Fig. 4(d), and a typical example ofa weakly unstable GS, which keeps a nearly undisturbed shape, is presented in Fig. 5. For ǫ = 6, the stable part of FIG. 2: (Color online) (a) A typical example of stable on-site-centered symmetric GSs, found in the first finite bandgap, at µ = − b = 0, and ǫ = 6. Here, and in similar figures below, the background pattern (red sinusoid) represents the scaledunderlying Rabi lattice [periodic modulation of the coupling constant, − ( ǫ/
6) cos (2 x ), with scaling factor 1 / µ = 3 and ǫ = 6. As seen in panels (c) and (d), development of the oscillatory instability replaces the GS by a“turbulent” pattern, which remains spatially confined. the first finite bandgap is − . < µ < .
14 and 0 < N < .
11 in terms of the chemical potential and total norm,respectively.In the present system the GSs are primarily unstable in the second finite bandgap, as is indicated in Fig. 4(b), andillustrated by a typical example in Fig. 3, which shows that the unstable solitons evolve into spatially confined chaoticmodes (“solitons of conservative turbulence”). An exception is the green dotted segment, which contains alternatingstable and unstable solitons, as shown in detail in Fig. 4(c).
FIG. 4: (Color online) (a) The numerically found amplitude of fundamental on-site-centered symmetric GSs versus chemicalpotential µ , for ǫ = 6 and b = 0. The dashed magenta curve is produced by the Thomas-Fermi approximation (TFA) for thesingle-component GSs: U ( x = 0) = V ( x = 0) = √ µ − µ ) − / . Panel (b) shows the total norm N versus µ , along withits dashed-magenta TFA counterpart, N = 2 hp − µ + µ cos − ( − µ/ i . In these panels, as well as in Fig. 7(a) below, redsolid and black dashed segments represent stable and unstable GSs, respectively, while the green dotted segment designatesa region of alternate stability and instability. The red dotted segment near the right edge of the first bandgap designates aregion of alternating stable GSs and ones weakly unstable against oscillatory perturbations. (c) The alternation of the stabilityand instability in the green dotted segment of (b). Here, red triangles and black circles represents stable and unstable GSs,respectively. (d) The alternation of the stability and weak oscillatory instability in the red dotted segment of (b). Here, andalso in Fig. 7(c) below, red triangles and blue filled circles represent stable GSs and ones subject to the weak instability,respectively. B. Off-site-centered spatially-even antisymmetric gap solitons
For antisymmetric states, with U ( x ) = − V ( x ) (in the system with b = 0), the above-mentioned effective single-component lattice potential, generated by the Rabi lattice, inverts its sign, taking the form of ǫ cos (2 x ). Accordingly,off-site-centered antisymmetric GSs tend to minimize their energy by placing two density maxima at potential-minimapoints, x = ± π/
2, which gives such solitons a chance to be stable. A typical example of a stable antisymmetric GSwith the off-site-centered spatially-even shape is shown in Fig. 6. The stability of this GS species is summarized inFig. 7(a), which demonstrates that they are stable solely in the first finite bandgap [the instability in the secondfinite bandgap is illustrated by Fig. 6(d)]. Similar to the situation shown for the on-site-centered symmetric GSs inFig. 4(b), at the right edge of the bandgap there is a narrow segment of alternating stability and weak oscillatoryinstability, whose structure is displayed in detail in Fig. 7(c).
C. Other types of symmetric and antisymmetric gap solitons
The system with b = 0 supports off-site-centered antisymmetric GSs with the spatially-odd shape, in additionto their even counterparts considered above. An example, and the family of such solitons in the ( µ, N ) plane, aredisplayed in Figs. 8 and 7(b), respectively. The conclusion of the analysis is that the family is completely unstablein both first and second finite bandgaps. Further, direct simulations demonstrate that, in the first bandgap, theinstability transforms the stationary spatially-odd GSs into persistent breathers, see Fig. 8(c), while in the secondbandgap, the unstable GSs evolve into apparently turbulent spatially confined modes, as shown in Fig. 8(d). In eithercase, the dynamical states produced by the instability keep the spatially odd shape, as indicated by the persistence FIG. 5: (Color online) An example of a weakly unstable GS with µ = 0 .
32, belonging to the red dotted segment near the rightedge of the first finite bandgap in Fig. 4. (a) Unstable eigenvalues of small perturbations are complex with small imaginaryparts. (b,c) Direct simulations corroborate the weak oscillatory instability of the soliton, which keeps its localized shape. (d)The weak instability is additionally illustrated by the time dependence of squared amplitudes of both components of the samesolutions.FIG. 6: (Color online) Panels (a) and (b) display a typical stable off-site-centered antisymmetric spatially-even GS for µ = − µ = 2, demonstrate that this unstable soliton is finallytransformed into a spatially confined turbulent state. of zero amplitude at the midpoint in Figs. 8(c) and 8(d).Additional types of solitons have been found too: on-site-centered antisymmetric, off-site-centered spatially-evensymmetric, and off-site-centered spatially-odd symmetric GSs. Due to the nature of the present system, in which theGSs are supported by the Rabi lattice, these additional species are actually tantamount to the three species consideredabove. Indeed, the on-site-centered symmetric GS is obviously equivalent to an antisymmetric one which is centeredat the site shifted by half a spatial period, their stability being identical too. The same pertains to off-site-centered FIG. 7: (Color online) (a) Total norm N versus chemical potential µ for off-site-centered antisymmetric spatially-even GSs,cf. Fig. 4(b) for the on-site centered symmetric solitons. (b) The N ( µ ) curve for the family of off-site-centered antisymmetricspatially-odd GSs. Here, the red and black dashed curves refer to unstable solitons which evolve, respectively, into robustbreathers [see Fig.8(c)], or into a confined turbulent state shown in Fig.8(d). (c) The detailed structure of the red-dottedsegment with alternate stability and instability in panel (a). The red triangles and blue filled circles label stable and weaklyunstable solutions, respectively.FIG. 8: (Color online) (a,b) An example of an unstable off-site-centered spatially-odd antisymmetric GS for µ = − b = 0 and ǫ = 6. (c) The perturbed evolution (with initial random perturbations at the5% level) of the same soliton shows its transformation into a persistent breather. (d) Simulations of the perturbed evolution ofthe GS of the same type, but with µ = 2 (which falls into the second finite bandgap) show its transformation into a confinedturbulent mode. FIG. 9: (Color online) Examples of stable on-site-centered asymmetric GSs with ǫ = 6, and fixed chemical potential µ = − . b = 0 .
2; (b) b = 1; (c) b = 3. Fields U ( x ) and V ( x ) with larger and smaller amplitudes are shown by blue and green lines,respectively. spatially-even and odd symmetric GSs, which may be easily converted into their antisymmetric counterparts. IV. ON-SITE-CENTERED GAP SOLITONS IN THE ASYMMETRIC SYSTEMA. Numerical results
It is relevant to stress that, while GSs in symmetric dual-core systems with the usual lattice potential and constantinter-core coupling readily feature spontaneous breaking of the (anti-)symmetry between their components, followedby generation of asymmetric solitons, provided that the nonlinearity strength exceeds a certain critical value [20, 21],this effect is not observed in the present system, i.e., all the GSs, both stable and unstable ones, are either symmetricor antisymmetric with respect to the two components. In this section, we report results for the most fundamentalon-site-centered asymmetric GSs, which are naturally produced by the asymmetric system, based on Eqs. (1), (2)with b = 0 (obviously, it is sufficient to consider b > b , the shapes of the U ( x ) and V ( x ) components become less localized and develop undulations in their tails. Examples of the perturbed evolutionof stable and unstable asymmetric GSs are further shown in Fig. 10. Different from the case of symmetric on-site-centered GSs, asymmetric ones which are unstable tend to develop a chaotic state expanding to the entire spatialdomain, cf. Fig. 3(c,d)Results obtained for families of asymmetric GSs and their stability are summarized in Fig. 11, by means of N ( µ )and R ( N ) curves [recall R is the asymmetry ratio defined in Eq. (9)]. These dependences suggest that the asymmetricGSs are more stable at lower values of the total norm, N , and stability regions shrink with the increase of asymmetrycoefficient b and asymmetry ratio R . Similar to the symmetric system ( b = 0), here there also exist regions of alternatestability and instability, which are designated by dotted segments in Fig. 11. The detailed structure of these segmentsis rendered in Fig. 12.In Fig. 13, we have collected results produced by the stability analysis for the on-site asymmetric GSs in the planesof ( b, µ ) and ( N, b ). Figures 13(a,b) demonstrate that the stability area originally shrinks with the increase of theasymmetry coefficient, b , and then stays narrow but nearly constant. In the ( N, b ) plane, the stability region alsonarrows at first with the increase of b , but then it broadens at still larger b .0 FIG. 10: (Color online) (a,b) Direct simulations of the perturbed evolution of a stable asymmetric on-site-centered GSs with( b, µ ) = (3 , − . b, µ ) = (3 , − . N of asymmetric on-site-centered GSs versus chemical potential µ , at ǫ = 6 and fixedvalues of the asymmetry coefficient: b = 1 , b = 3, and b = 5. (b) Asymmetry ratio, R , defined as per Eq. (9), versus N , forthe same soliton families. The family with b = 0, which has R ≡
0, is included too, for the completeness’ sake. In these panels,solid and dashed lines represent stable and unstable GSs, respectively, while the dotted segments designate regions of alternatestability and instability.
B. A semi-analytical approximation
In the limit of b ≫
1, the two-component system, based on Eqs. (1) and (2) can be easily reduced to a singleequation with the usual lattice potential, by means of an approximation similar to that recently elaborated for thetwo-component BEC under the action of strong Zeeman splitting in Ref. [37]. The approximation is based on thefact that large b gives rise to solutions with chemical potential µ = − b + δµ, which implies | δµ | ≪ b . Accordingly,the solutions are looked for in the form of { u ( x, t ) , v ( x, t ) } = exp ( ibt ) { ˜ u ( x, t ) , ˜ v ( x, t ) } , where the remaining t -dependence in ˜ u and ˜ v is slow, in comparison with exp( ibt ). Then Eq. (2), in which the the XPM terms may berestored, as per Eq. (3), readily yields an approximate expression for the small v component:˜ v ≈ ǫ cos (2 x )2 b + σg | u | ˜ u ≈ ǫ (cid:18) b − σg b | u | (cid:19) cos(2 x ) · ˜ u. (12)To justify the expansion of the the fraction in this expression, it is assumed that the soliton’s peak density is not toolarge, viz ., (cid:0) | u | (cid:1) max ≪ b/ | g | (13)1 FIG. 12: (Color online) Panels (a), (b), and (c) show the detailed structure of regions with alternating stable and unstableon-site-centered asymmetric GSs in the regions shown by the dotted line in Fig. 11(a), for b = 1 , , and 5, respectively. Redtriangles and black circles represent, severally, stable and unstable solitons.FIG. 13: (Color online) Stability diagram for on-site-centered asymmetric GSs in the planes of ( b, µ ) (a) and ( N, b ) (b). Thered and gray colors designate, respectively, the stability area, and the one of alternate stability and instability. GSs do not existin the top blank areas in panels (a). In the bottom blank area in (a) and blank area in (b), there exists completely unstableGSs. (this condition is easily satisfied for large b ). Then, the substitution of this approximation in Eq. (1) leads to a singleequation for ˜ u ( x, t ), with an effective lattice potential: (cid:20) i ∂∂t + 12 ∂ ∂x − σ (cid:18) ǫ g b (cid:19) | ˜ u | + ǫ b cos (2 x ) (cid:21) ˜ u = 0 (14)[it is easy to see that, to obtain the nonlinearity coefficient in Eq. (14) under the above condition (13), one maysubstitute cos (2 x ) by its average value, 1 / , while this substitution is not relevant in the effective lattice potential].In turn, Eq. (14) with σ (cid:0) ǫ g/ b (cid:1) > FIG. 14: (Color online) Panels (a1,a2) and (b1,b2) present typical examples of stable GSs produced, respectively, by thefull system of Eqs. (1) and (2), and by the semi-analytical approximation which amounts to the single equation (14), for( b, µ ) = (10 , − . u component, in panels (a3) and (b3), respectively. coefficient, g , is essential if the Rabi lattice is strong enough, namely, | g | ǫ ∼ b , which is compatible with theunderlying condition b ≫ V. CONCLUSION
The objective of this work is to introduce the model of two-component gap solitons, based on two GPEs (Gross-Pitaevskii equations) with self-repulsive nonlinearity, coupled by linear terms which are subject to the spatiallyperiodic cosinusoidal modulation (
Rabi lattice ). The system can be implemented in binary BEC with the superimposedstanding wave of a resonantly-coupling electromagnetic field, and in the bimodal light propagation in twisted fibers.We have demonstrated that this setting gives rise to stable two-component GSs (gap solitons), in the absence ofperiodic potentials which are necessary for the existence of GSs in usual models. Several types of the GSs havebeen found, including on-site-centered symmetric and antisymmetric modes, and spatially even and odd off-site-centered symmetric and antisymmetric ones. Both symmetric and antisymmetric GSs are stable chiefly in the firstfinite bandgap, as well as in a small segment of the second bandgap. These findings are, roughly, similar to whatwas recently found for the stability of usual GSs in the single-component model with a periodic potential [36]. A3noteworthy finding is the alternation of stable and unstable on-site-centered symmetric GSs in the latter segment. Foron-site-centered symmetric and off-site-centered spatially even symmetric and antisymmetric GSs, there also existsa narrow segment of alternating stability and weak oscillatory instability near the right edge of first finite bandgap.Unstable GSs spontaneously transform into robust breathers or spatially confined turbulent states. On-site-centeredGSs were found in the asymmetric system too, where segments featuring the alternate stability exist as well. Thus, thealternation of stability and instability of GSs, which was not reported in previously studied models, is a characteristicgeneric feature of the present system. It is worthy to note that the stability area of the on-site-centered asymmetricGSs originally shrinks with the increase of the asymmetry coefficient, b , but then it expands with the further increaseof b , in terms of the total power of the solitons. In the limit of b ≫
1, an analytical approximation makes it possibleto transform the system into a single GPE with an effective periodic potential and respective GS solutions.A natural extension of the present analysis should produce a detailed analysis of the system including the XPMinteraction between the two components. A challenging direction for further work is a two-dimensional version ofthe present system. In that case, two-component solitary vortices may be looked for, in addition to fundamentalGSs. In fact, off-site-centered spatially odd GSs, which are considered above, are one-dimensional counterparts of thetwo-dimensional vortices.
Acknowledgments
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