Lossless Polariton Solitons
LLossless Polariton Solitons
Stavros Komineas ∗ , Stephen P. Shipman † and Stephanos Venakides ‡ ∗ Department of Mathematics and Applied MathematicsUniversity of CreteHeraklion, Crete, Greece † Department of MathematicsLouisiana State UniversityBaton Rouge, Louisiana 70803, USA ‡ Department of MathematicsDuke UniversityDurham, North Carolina 27708, USA
Abstract.
Photons and excitons in a semiconductor microcavity interact toform exciton-polariton condensates. These are governed by a nonlinear quantum-mechanical system involving exciton and photon wavefunctions. We calculate allnon-traveling harmonic soliton solutions for the one-dimensional lossless system.There are two frequency bands of bright solitons when the inter-exciton interac-tions produce an attractive nonlinearity and two frequency bands of dark solitonswhen the nonlinearity is repulsive. In addition, there are two frequency bands forwhich the exciton wavefunction is discontinuous at its symmetry point, where itundergoes a phase jump of π . A band of continuous dark solitons merges with aband of discontinuous dark solitons, forming a larger band over which the solitonfar-field amplitude varies from 0 to ∞ ; the discontinuity is initiated when the op-erating frequency exceeds the free exciton frequency. The far fields of the solitonsin the lowest and highest frequency bands (one discontinuous and one continuousdark) are linearly unstable, whereas the other four bands have linearly stable farfields, including the merged band of dark solitons. Key words: polariton, soliton, exciton, photon, nonlinear, semiconductor microcavity
Exciton-polaritons are a quantum-mechanical quasiparticle formed by the coupling of pho-tons with excitons. An exciton is a dipole generated in a semiconductor when an electronabsorbs a photon and jumps from the valence to the conduction band thus leaving a hole inthe valence band. The electron and hole are attracted to each other by an effective electro-static Coulomb force, resulting in the excitation of an electron-hole pair. Exciton-polaritons1 a r X i v : . [ n li n . PS ] J u l an be trapped in a planar microcavity containing a semiconductor material, that is, theylive in a two-dimensional quantum well.Exciton-polaritons can form Bose-Einstein condensates (BEC) at relatively high temper-atures [4, 5, 12, 14], sustained by continuous laser pumping of photons. The condensatewavefunctions produce a rich variety of localised quantum states in the micrometer scale:dark solitons [2, 10, 11, 16, 20], bright solitons [7, 8, 16, 22], and vortices [9, 17]. Solitons inpolaritonic condensates have potential for applications in ultrafast information processing [1]due to picosecond response times and strong nonlinearities [8, 22]. See [23], for example, fora tour of polariton condensates.In a mean-field approximation, the excitons and photons are described by separate wave-functions ψ X (excitons) and ψ C (photons) of spatial coordinates x = ( x , x ) and time t .A continuous absorption and emission of photons by atoms in the semiconductor (Rabi os-cillation) is represented by a coupling of the two equations. The kinetic term (Laplacian)is typically neglected for the excitons due to their significantly larger mass. On the otherhand, exciton-exciton interaction is significant, so a nonlinear term arises in the equation forthe exciton field. The system of equations reads [3, 6, 13, 18, 25] i∂ t (cid:32) ψ X ψ C (cid:33) = (cid:32) ω X − iκ X + g | ψ X | γγ ω C − iκ C − (cid:126) m C (cid:52) (cid:33) (cid:32) ψ X ψ C (cid:33) + (cid:32) F (cid:33) . (1.1)The numbers ω X,C and κ X,C are real; ω X is the frequency of a free exciton, ω C is thefrequency of the free, zero-momentum photon; κ X and κ C are the attenuation constants ofthe exciton and photon and account for losses; m C is the mass associated with the photons.The Laplacian is denoted by (cid:52) = ∂ x + ∂ x . The forcing F represents a pumping of photonsinto the microcavity. The coupling associated with the Rabi oscillations enters throughthe frequency parameter γ , which is half the Rabi frequency. The nonlinearity g | ψ X | isattractive when g < g > κ X and κ C are zero) and unforced ( F = 0). In turning off bothpumping and losses, which are due to radiation and thermalization, we focus on the synergyof exciton interaction (nonlinearity) and photon dispersion. We consider fields that dependon only one spatial variable, say x , and we use the notation x = x below. Under these con-ditions, we discover and analyze three families solitons that exhaust all harmonic stationary(non-traveling) exciton-polariton solitons. We use the term “soliton” in a broad sense torefer to a field with amplitude that tends to a constant value as | x | → ∞ , which is typicalin the physics literature. All solitons can be expressed exactly by quadrature through exactintegration of the harmonic polariton system (see (3.52)–(3.53)).For each of the three families, the solitons exist on two frequency bands, all six bandsbeing mutually disjoint (Fig. 1). There is one family of dark solitons for g >
0, one family ofbright solitons for g <
0, and one family of solitons whose exciton wavefunction exhibits aspatial jump discontinuity. The discontinuity is made physically possible by the vanishing ofthe photon field, which brings dispersion to the system, at the point of discontinuity of theexciton field; mathematically, these are distributional solutions of the polariton equations.2 C ⇣ = ( ! ! X )( ! ! C ) dark g > g > g < g < g > g < ! ! ⇣ ! ! LP ! UP ! ! LP ! UP ! X ! X linear homogeneous polariton bandsnonlinear polariton soliton bandslower band upper band ! C Figure 1:
The lossless, unforced, one-dimensional polariton equations admit six frequency bandsof stationary soliton-type solutions, whose graphs are shown in Fig. 2 and in section 3.2. Positivedetuning ω C > ω X is shown here; for negative detuning ω C < ω X , band 3.2 is absent (Fig. 8).Solitons in bands 1.1, 1.2, and 3.2 are all dark and can coexist in a system with g >
0. Solitonsin bands 2.1, 2.2, and 3.1 can coexist in a system with g <
0. The linear stability of the far-fieldvalue of the soliton as a constant-amplitude solution of the polariton equations is indicated. Band1.1 = ( ω , ω X ) coincides with the lower band of homogeneous linear solitons when ω C > ω X , andband 1.2 begins at the minimal value of the upper band ( ω , ∞ ) of homogeneous linear solitons.The endpoint frequencies of the bands are defined as follows. Set p ( ω ) := ( ω − ω X )( ω − ω C ), with ω X and ω C defined after (1.1), p ( ω ) = p ( ω ) = γ and p ( ω ) = γ and p ( ω ) = p ( ω ) = γ . The stationary dark solitons in bands 1.1 and 3.2 were introduced in [15]. The presentwork exhausts all soliton solutions and thus proves that these two bands (collectively calledband D) are the only dark stationary 1D solitons with stable far-field values. When ω C >ω X (positive detuning), band 1.1 coincides with the “lower polariton” band of linear ( g =0) homogeneous polaritons . The upper band 1.2 of dark solitons has the same minimalfrequency as the “upper polariton” band of homogeneous linear solitons. The dispersionrelations for the lower and upper bands of linear homogeneous polaritons are shown in [18,Fig. 1]. 3 and 1.2: g > g < g < g < g > g > | X | | X | | X | | X | | X | | X | x x xxxx Figure 2:
The square modulus | ψ X ( x ) | of the exciton field of the solitons in the bands depictedin Fig. 1. The exciton fields ψ X ( x ) of the solitons in bands 3.1 and 3.2 are antisymmetric andexperience a discontinuity at the point where | ψ X ( x ) | has a v-shape (see Fig. 6) (note that thesoliton of band 1.1 is smooth, although sharp, at its nadir). The far-field asymptotic value of thesolitons in bands 1.1, 1.2, 3.1, and 3.2 is ζ ∞ /g (defined in Eq. (2.20)). Bands 1.1 and 3.2 are unifiedinto one band, as described in section 3.3. We may interpret the solitons derived in this work as ideal analytic descriptions of lo-calized polariton formations in high-Q microcavities (see, e.g. , [19]). It is plausible that,within the finite region of a physical microcavity, the far-field values may be maintained bya small pumping. Ref. [24] reports the creation of polariton condensates at two pump spots,and localized structures can be sustained in the region between the two spots where there isno pumping. In [2, 10], quasi-one-dimensional polariton structures are observed outside thepump spots.
Consider a lossless, unforced, one-dimensional polariton field, consisting of a photon wave-function ψ C ( x, t ) and an exciton wavefunction ψ X ( x, t ) dynamically coupled through theirstandard quantum-mechanical equations, i∂ t ψ X = (cid:0) ω X + g | ψ X | (cid:1) ψ X + γψ C , (2.2) i∂ t ψ C = (cid:0) ω C − ∂ xx (cid:1) ψ C + γψ X . (2.3)obtained by restricting (1.1) to one spatial dimension and setting κ X,C = 0 and F = 0. Thetime variable t is normalized to an arbitrary unit of time T , frequencies (including ω X , ω C , γ , and g ) are normalized to 1 /T , and the spatial variable x is normalized to (cid:112) T (cid:126) /m C . Thusall variables and parameters are non-dimensional.4he polariton equations (2.2,2.3) admit two quantities that are conserved in time, N = (cid:90) (cid:0) | ψ X | + | ψ C | (cid:1) dx, (2.4) H = (cid:90) (cid:0) | ∂ x ψ C | + ω C | ψ C | + ω X | ψ X | + g | ψ X | + 2 γ Re( ψ X ψ C ∗ ) (cid:1) dx. (2.5)A traveling-wave polariton field with carrier frequency ω , modulated by an envelope hasthe form ψ X ( x, t ) = φ X ( x − ct ) e i ( kx − ωt ) , (2.6) ψ C ( x, t ) = φ C ( x − ct ) e i ( kx − ωt ) . (2.7)Under this ansatz, the polariton equations are equivalent to the pair − icφ (cid:48) X = (cid:0) ω X − ω + g | φ X | (cid:1) φ X + γφ C , (2.8) i ( k − c ) φ (cid:48) C = (cid:16) ω C − ω + k (cid:17) φ C − φ (cid:48)(cid:48) C + γφ X , (2.9)in which the prime denotes the derivative with respect to the argument.This system of two complex ODEs reduces to a system of two real ODEs when thepolariton envelope depends only on the spatial variable (speed of travel c = 0) and thepolariton carrier phase is spatially invariant (wavenumber k = 0):( ψ X ( x, t ) , ψ C ( x, t )) = ( φ X ( x ) , φ C ( x )) e − iωt . (2.10)Under this assumption, and with the notation (cid:36) X = ω − ω X , (cid:36) C = ω − ω C , the pair of real functions ( φ X , φ C ) satisfies the equations (cid:0) gφ X − (cid:36) X (cid:1) φ X + γφ C = 0 , (2.11) − φ (cid:48)(cid:48) C − (cid:36) C φ C + γφ X = 0 . (2.12)The first equation fully determines the photon field φ C as an odd cubic polynomial functionof the exciton field φ X , illustrated in Fig. 3. Thus the field value pair ( φ X ( x ) , φ C ( x )) runsalong the graph of the cubic as the spatial variable x varies. For the solitons in bands 1.1,1.2, 3.1, and 3.2, the field pair lies on this cubic between the two nonzero equilibrium points( φ ∞ X , φ ∞ C ) and ( − φ ∞ X , − φ ∞ C ) of the system of equations (2.11,2.12), where φ ∞ X = (cid:115) g (cid:18) (cid:36) X − γ (cid:36) C (cid:19) ,φ ∞ C = γ(cid:36) C φ ∞ X . (2.13)5 X < (cid:36) X > g < X C X C g > X C X C Figure 3:
The relation (2.22) that gives the photon envelope value φ C vs. the exciton envelopevalue φ X of a harmonic solution of the form ( ψ X ( x, t ) , ψ C ( x, t )) = ( φ X ( x ) , φ C ( x )) e − iωt of thepolariton equations (2.2,2.3). Its shape depends on the signs of g and (cid:36) X = ω − ω X . A solitonsolution cannot cross the vertical dotted lines through the critical points. These equilibrium points are indicated by open dots in the graphs in section 3.2. Theycorrespond to homogeneous (spatially constant and time-harmonic) solutions of the polaritonequations (2.2,2.3), ( ψ X ( x, t ) , ψ C ( x, t )) = ± ( φ ∞ X , φ ∞ C ) e − iωt . (2.14)In the case that g(cid:36) X <
0, equation (2.11) defines a monotonic relation between φ C and φ X (the off-diagonal graphs in Fig. 3). Thus (2.12) can be written in the form φ (cid:48)(cid:48) C ( x ) + U (cid:48) ( φ C ( x )) = 0 , (2.15)which results in the conservation of an energy-type function, ( φ (cid:48) C ) + U ( φ C ) = K, (2.16)with K being an arbitrary constant. This equation describes orbits of the system in the( φ C , φ (cid:48) C )-plane (phase plane); homoclinic and heteroclinic orbits correspond to solitons. Inthe case that g(cid:36) X >
0, the relation (2.11) between φ C and φ X has three monotonic branches(the diagonal graphs in Fig. 3) separated by two critical points, where φ C achieves a localmaximum or minimum as a function of φ X . These two critical points occur when gφ X = (cid:36) X / . (2.17)Within the domain of each branch separately, an equation of the form (2.16) holds.The system (2.11,2.12) can in fact admit a solution that passes through either criticalpoint, from one branch of the cubic (2.11) to another. But such a solution is unique—thereis no arbitrary constant of integration analogous to the constant K in (2.16). Moreover, asolution passing through a critical point cannot be a soliton; it is either periodic or becomesunbounded as | x | → ∞ . This is stated in part (c) of Theorem 1.One therefore knows that all soliton solutions of (2.11,2.12) are confined to a singlemonotonic branch of the cubic (2.11)—the exciton envelope function φ X ( x ) cannot passthrough the values (cid:112) (cid:36) X / (3 g ). We will refer to this as the connectivity condition for polaritonsolitons. 6nalysis of all soliton solutions of (2.11,2.12), especially with regard to their dependenceon (cid:36) X , (cid:36) C , and g , is complex. We find that working with the variable ζ = gφ X (2.18)renders the analysis most transparent. The system (2.11,2.12), once integrated, becomes afirst-order ODE (Theorem 1) that expresses ( ζ (cid:48) ) as a rational function of ζ , in which thenonlinearity parameter g is no longer present. The forbidden value ζ = (cid:36) X /
3, occurring atthe critical points of the cubic (2.11), is manifest as a singularity of the equation( ζ (cid:48) ) = 89 ζQ ( ζ ) (cid:0) ζ − (cid:36) X (cid:1) . (2.19)( Q is a cubic polynomial defined below.) The sign of g is determined by the sign of ζ ( x ),which is constant for any solution.In terms of ζ , the equilibrium points of (2.11,2.12) are expressed as g ( φ ∞ X ) = ζ ∞ , where ζ ∞ is the non-dimensional frequency ζ ∞ = ζ ∞ ( ω ) := (cid:36) X − γ (cid:36) C = ω − ω X − γ ω − ω C . (2.20) Theorem 1. (a) The pair of equations (2.11,2.12) implies the pair (3 ζ − (cid:36) X ) ζ (cid:48) = 8 ζ Q ( ζ ) , (2.21) φ C = γ ( (cid:36) X − gφ X ) φ X , (2.22) in which ζ ( x ) = gφ X ( x ) and Q ( ζ ) is a cubic polynomial in ζ , Q ( ζ ) = Q ( ζ ; ω ) = − (cid:36) C (cid:0) ζ − (3 ζ ∞ + (cid:36) X ) ζ + ζ ∞ (cid:36) X ζ + K (cid:1) (2.23) and K is a constant of integration. Conversely, whenever φ (cid:48) X ( x ) φ (cid:48) C ( x ) (cid:54) = 0 , the pair (2.21,2.22)implies the pair (2.11,2.12).(b) Whenever g and (cid:36) X have the same sign, the local extrema of φ C = γ − ( (cid:36) X − gφ X ) φ X occur when ζ = gφ X = (cid:36) X / . The roots of Q (cid:48) are ζ ∞ and (cid:36) X / . Thus, whenever (cid:36) X / isa root of Q , the factor (3 ζ − (cid:36) X ) appears on both sides of (2.21).(c) If a solution ( φ X ( x ) , φ C ( x )) of (2.11,2.12) passes through a critical point of the cubic(2.11), then either the solution is periodic or | φ C ( x ) | and | φ X ( x ) | tend to ∞ as | x | → ∞ .Proof. To prove these statements, the structure of equations (2.11,2.12) is illuminated bywriting them as f ( φ X ) + γφ C = 0 , (2.24) f ( φ C ) + γφ X = φ (cid:48)(cid:48) C , (2.25)in which f and f are odd polynomials. 7y multiplying the first equation by φ (cid:48) X and the second by φ (cid:48) C , adding, and then takingantiderivatives, one obtains K (cid:48) + ˜ f ( φ X ) + ˜ f ( φ C ) + γφ X φ C = ( φ (cid:48) C ) , (2.26)in which the even polynomials ˜ f , are primitives of f , and K (cid:48) is an arbitrary constant.Equation (2.24) expresses φ C as an odd polynomial function of φ X , so the left-hand side of(2.26) is an even polynomial function of φ X , say P ( φ X ). Thus (2.24,2.25) is equivalent tothe validity of the pair P ( φ X ) = ( φ (cid:48) C ) , (2.27) f ( φ X ) + γφ C = 0 , (2.28)for some constant K (cid:48) in the definition of P , which is computed to be P ( φ ) = K (cid:48) − (cid:36) C g γ φ + g (cid:18) (cid:36) X (cid:36) C γ − (cid:19) φ + (cid:36) X (cid:18) − (cid:36) X (cid:36) C γ (cid:19) φ . (2.29)Equation (2.27) can be written equivalently in terms of φ X alone by differentiating (2.28)with respect to x and substituting the resulting expression for φ (cid:48) C into the right-hand sideof (2.27), 4 γ P ( φ X ) = f (cid:48) ( φ X ) ( φ (cid:48) X ) . (2.30)Since P is even, it is a cubic polynomial function of ζ = gφ X , and a calculation converts(2.30) into the differential equation stated in the theorem,(3 ζ − (cid:36) X ) ζ (cid:48) = 8 ζ Q ( ζ ) , (2.31)in which the cubic polynomial Q is related to P through 2 gγ P ( φ X ) = Q ( ζ ) and is given by Q ( ζ ) = 2 gγ K (cid:48) + ζ(cid:36) X ( γ − (cid:36) X (cid:36) C ) − ζ (cid:0) γ − (cid:36) X (cid:36) C (cid:1) − ζ (cid:36) C . (2.32)Notice that g has disappeared from the differential equation (2.21) except where it is multi-plied by the constant of integration in Q .We must now show that Q has the form given in part (a) of the Theorem and provepart (b). For part (b), differentiate (2.27) with respect to x to obtain P (cid:48) ( φ X ) φ (cid:48) X = φ (cid:48)(cid:48) C φ (cid:48) C . (2.33)Then, by using (2.25) and the x -derivative of (2.28), equation (2.33) is rewritten as P (cid:48) ( φ X ) = − f (cid:48) ( φ X ) (cid:0) φ X + γ − f ( φ C ) (cid:1) (2.34)when φ (cid:48) X (cid:54) = 0. But both sides of this equation are polynomials in φ X (recall φ C is a polynomialfunction of φ X ), so this leads to the observation that any root of the polynomial f (cid:48) is also aroot of P (cid:48) . But f (cid:48) ( φ X ) = 3 gφ X − (cid:36) X , so P (cid:48) ( φ X ) = 0 when gφ X = (cid:36) X / . (2.35)8rom the relation 2 gγ P ( φ X ) = Q ( ζ ), one finds that Q (cid:48) ( (cid:36) X /
3) = 0 (if (cid:36) X (cid:54) = 0) , (2.36)which proves part (b).The other root of Q (cid:48) is found to be ζ ∞ = (cid:36) X − γ (cid:36) C , and one can write Q as stated inthe theorem (2.23), with K = − gγ K (cid:48) (cid:36) C . (2.37)To prove part (c), suppose first that a classical solution of (2.11,2.12) satisfies ζ (0) = gφ X (0) = (cid:36) X /
3, so that the first factor of the left-hand side of (2.21) vanishes at x = 0.Thus Q ( (cid:36) X /
3) = 0, and by part (b), (cid:36) X / Q , and (2.21) reduces to( ζ (cid:48) ) = − (cid:36) C ζ ( ζ − ζ ) , (2.38)in which ζ = 4 (cid:36) X − γ (cid:36) C . By standard phase-plane analysis, the solutions of this equation are either constant, period-ically oscillating between ζ = 0 and ζ = ζ , or tending to ∞ as | x | → ∞ .Next, suppose that ( (cid:36) X / Q ( (cid:36) X / > ζ ( x ) is continuous with ζ (0) = (cid:36) X / h i ( y ) are analytic at y = 0 with h i (0) >
0. Because the numerator in the differential equation ζ (cid:48) = ± (cid:112) ζQ ( ζ )3 ζ − (cid:36) X (2.39)is positive at ζ = (cid:36) X /
3, one obtains, for x near 0, either ζ ( x ) = (cid:36) X | x | / h ( | x | / ) (2.40)or ζ ( x ) = (cid:36) X − | x | / h ( −| x | / ) , (2.41)in which one sign is chosen for x > x <
0. Denote˜ φ X ( x ) = φ X ( x ) − φ X (0) , ˜ φ C ( x ) = φ C ( x ) − φ C (0) , (2.42)with φ X (0) = ± (cid:112) (cid:36) X / (3 g ) . From (cid:36) X (cid:54) = 0 and the equation φ X ( x ) = (cid:112) ζ ( x ) /g (the sign of g is chosen to make the square root real), one obtains˜ φ X ( x ) = | x | / h ( | x | / ) or − | x | / h ( −| x | / ) . (2.43)Since ( φ X (0) , φ C (0)) is a critical point of the cubic φ C vs. φ X relation, one has˜ φ C ( x ) = ± ˜ φ X ( x ) h ( ˜ φ X ( x )) = ±| x | h ( ±| x | / ) . (2.44)It follows that φ (cid:48)(cid:48) C ( x ) has a leading singular part equal to a nonzero multiple of the deltafunction δ ( x ), and this is inconsistent with equation (2.12).9 Polariton solitons
A solution of the ODE (2.21) corresponds to a stationary, or non-traveling, time-harmonicsolution of the polariton system (2.2,2.3). Our interest is in soliton solutions, for which thespatial envelope has a limiting value as | x | → ∞ . All solitons and their frequency bands aredescribed in Theorem 2 below, and proved in section 5. The system admits bands of darkand bright solitons.In addition, we find solitons for which the exciton field is discontinuous at its point ofsymmetry where the photon field vanishes. These are distributional solutions of the solitonequations; this is proved in section 5 (p. 24). The physical origin of the discontinuity isthat the vanishing of the photon field at a point in space turns off the interaction betweenneighboring excitons because this interaction is mediated only by the coupling of the excitonfield to the dispersive photon field. When ω X < ω C , a band of continuous dark solitons anda band of discontinuous dark solitons can be unified into a single band of dark solitons, asdescribed in section 3.3. The reduction of the polariton system (2.2,2.3) to an ODE (2.21) under the assumptionof harmonic solutions allows a complete derivation of all stationary soliton solutions of(2.2,2.3). By a stationary soliton solution, we mean a harmonic solution ( ψ X ( x, t ) , ψ C ( x, t ))= ( φ X ( x ) , φ C ( x )) e − iωt for which the envelopes ( ψ X ( x ) , ψ C ( x )) tend to far-field values as x → ±∞ . The form of the ODE, ζ (cid:48) = f ( ζ ) guarantees that | ζ ( x ) | and therefore also | φ X ( x ) | exhibit a single maximum at the soliton peak or minimum at the soliton nadir. Theorem 2.
There are bright, dark, and discontinuous stationary soliton solutions of thelossless, unforced polariton equations (2.2,2.3) for frequencies within certain bands that de-pend on ω X , ω C , and γ . The three soliton classes described below exhaust all solutions of theform ( ψ X ( x, t ) , ψ C ( x, t )) = ( φ X ( x ) , φ C ( x )) e − iωt (3.45) for which φ X ( x ) and φ C ( x ) have limits (far-field values) as x → ±∞ .1. Dark solitons. (Red bands in Figs. 4 and 7) Equations (2.2,2.3), for g > , admitsolutions of the form (3.45) for which φ X ( x ) and φ C ( x ) are antisymmetric, monotonic,and bounded. The frequency bands for which these solutions exist are given by < (cid:36) X (cid:36) C < γ if ω < min { ω C , ω X } , (band 1.1) γ < (cid:36) X (cid:36) C < γ if ω > max { ω C , ω X } . (band 1.2) (3.46) The far-field (suprimal) value of | φ X ( x ) | is lim | x |→∞ | φ X ( x ) | = 1 √ g (cid:12)(cid:12)(cid:12)(cid:12) (cid:36) X − γ (cid:36) C (cid:12)(cid:12)(cid:12)(cid:12) / . (3.47)10 . Bright solitons. (Green bands in Figs. 5 and 7) Equations (2.2,2.3), for g < , admitsolutions of the form (3.45) for which φ X ( x ) and φ C ( x ) are symmetric and boundedand have a unique local maximum or minimum. The frequency bands for which thesesolutions exist are given by γ < (cid:36) X (cid:36) C < γ if ω < min { ω C , ω X } , (band 2.1) < (cid:36) X (cid:36) C < γ if ω > max { ω C , ω X } . (band 2.2) (3.48) These solitons vanish at the far field ( | x | → ∞ ), and | g | φ X ( x ) attains a maximalvalue of max −∞ 34 + (cid:114) − (cid:36) X (cid:36) C γ (cid:21) . (3.49) Discontinuous solitons. (Orange bands in Figs. 6 and 7) Equations (2.2,2.3) admitantisymmetric bounded solutions of the form (3.45) for which φ X ( x ) is discontinuous at x = 0 but φ C ( x ) is continuous. They satisfy the polariton equations in the distributionalsense, and away from the point of discontinuity they satisfy the equations classically.These solutions exist in the following two frequency bands.(a) For g < , and all frequencies satisfying ω < min { ω C , ω X } and (cid:36) X (cid:36) C > γ , (band 3.1)there is a soliton such that | φ X ( x ) | decreases monotonicallyfrom (cid:12)(cid:12)(cid:12)(cid:12) (cid:36) X g (cid:12)(cid:12)(cid:12)(cid:12) / down to (cid:112) | g | (cid:12)(cid:12)(cid:12)(cid:12) (cid:36) X − γ (cid:36) C (cid:12)(cid:12)(cid:12)(cid:12) / as x runs from the location of the peak of | φ X ( x ) | to ∞ . Where | φ X ( x ) | experiencesits peak, | φ C ( x ) | experiences its nadir.(b) For g > and all frequencies satisfying ω X < ω < ω C , (band 3.2)there is a dark soliton such that | φ X ( x ) | increases monotonicallyfrom (cid:12)(cid:12)(cid:12)(cid:12) (cid:36) X g (cid:12)(cid:12)(cid:12)(cid:12) / up to (cid:112) | g | (cid:12)(cid:12)(cid:12)(cid:12) (cid:36) X − γ (cid:36) C (cid:12)(cid:12)(cid:12)(cid:12) / as x runs from the location of the nadir of | φ X ( x ) | to ∞ . In the negative detuningcase, ω C < ω X , this band is absent. At the far field, the dark solitons and the discontinuous solitons tend to homogeneoussolutions (2.14) of the polariton equations, that is,lim | x |→∞ gφ X ( x ) = ζ ∞ ( ω ) , (3.50)11ith ζ ∞ given by (2.20). According to Theorem 1(b), ζ ∞ ( ω ) is one of the stationary pointsof Q ( ζ ; ω ) ( i.e. , ∂Q ( ζ ; ω ) /∂ζ = 0).The peak values of the bright soliton amplitudes are expressed through one of the rootsof Q ( ζ ) when K = 0 (as will be demonstrated in section 5), namely ζ ( ω ) := (cid:36) X − γ (cid:36) C (cid:104) + (cid:113) − (cid:36) X (cid:36) C γ (cid:105) . (3.51)Since g < ζ ( x ) is negative and, according to (3.49), attains a minimal value of ζ .Band 1.1 of dark solitons coincides with the lower band of linear homogeneous polaritons( ω , ω X ), and band 1.2 starts at the same minimal frequency ω as that of the upper bandof linear homogeneous polaritons [18, Fig. 1]. The frequencies ω and ω are the roots ofthe quadratic ( ω − ω X )( ω − ω C ) − γ , with ω < ω .Band 3.1 of discontinuous solitons for g < x = ± (cid:90) ζ ( x )0 z − (cid:36) X (cid:112) zQ ( z ) dz (0 ≤ ζ ( x ) ≤ ζ ∞ ) . (3.52)The exciton and photon fields are then obtained by φ X ( x ) = ± sgn( x ) (cid:115) ζ ( x ) g ,φ C ( x ) = γ ( (cid:36) X − gφ X ( x ) ) φ X ( x ) . (3.53)In band 3.2, expression (3.52) is modified by replacing the lower limit of integration by (cid:36) X and allowing (cid:36) X ≤ ζ ( x ) ≤ ζ ∞ . Similar expressions apply for the other soliton bands. The figures in this section depict the soliton solutions of the form (3.45) for the polaritonequations (2.2,2.3). The three types of solitons announced in Theorem 2 are depicted inthree separate figures below.In Figures 4, 5, and 6, assume that the symmetry point of each soliton (peak or nadir ofthe amplitude) is at x = 0. In each figure: (cid:114) The leftmost diagram shows two frequency bands of solitons on the ω -axis of the ωζ -plane.At a chosen frequency in each band, an arrow spans the range of ζ -values of a soliton,pointing toward the far-field value lim | x |→∞ ζ ( x ). The tail of the arrow, indicated by asolid dot, has its ordinate at ζ (0). The point of the arrow, indicated by an open circle,has its ordinate at the far-field value of ζ ( x ). (cid:114) The sign of g coincides with the sign of ζ = gφ X .12 The middle and rightmost diagrams depict the exciton and photon envelopes φ X ( x ) and φ C ( x ) for each frequency corresponding to the arrows in the leftmost diagram. (cid:114) The upper graphs depict the trajectory of the point ( φ X ( x ) , φ C ( x )) along the cubic relation(2.22) as x traverses the real line. The solid dots mark the central point ( φ X (0) , φ C (0)),and the open circles mark the far-field values lim x →±∞ ( φ X ( x ) , φ C ( x )). (cid:114) The lower graphs depict the exciton and photon envelopes vs. the spatial variable x . Whenone passes from ζ = gφ X to φ X , the extraction of square roots results in two solitons, whichare minuses of each other. One choice of square root is shown in the graphs. (cid:114) In each figure, γ = 1 and ω C − ω X = 1. The inequality ω C > ω X is referred to as “positivedetuning”. The “negative detuning” case ω C < ω X is depicted in Fig. 7 (right) and inFig. 8. - - - - - - - - - - - - - - ⇣ = ⇣ = ⇣ = ! ! X ! C X C X C X ( x ) C ( x ) X ( x ) C ( x ) x x ⇣ = g ( X ) Band 1.1 Band 1.2 ⇣ = ( ! ! X )( ! ! C ) Band 1.1 Band 1.2 ⇣ = ⇣ ( ! ) ⇣ = ⇣ ( ! ) Figure 4: Dark solitons. Anti-symmetric dark solitons for nonlinearity coefficient g > 0. (Seethe bullet points above in section 3.2 for a general explanation.) The far-field value ζ ∞ of ζ = gφ X ,indicated by the open dots in the leftmost diagram and given by (2.20), is equal to a double rootof the cubic Q ( ζ ) = Q ( ζ ; ω ) (see (2.23)) created by the appropriate choice of constant K = K ( ω ).In the upper graphs (middle and right), the pair ( φ X ( x ) , φ C ( x )) travels from one open circle to theother as x travels from −∞ to ∞ . - - - - - - - - - - - - - ⇣ = ⇣ = !! X ! C X C X C X ( x ) C ( x ) X ( x ) C ( x ) x x ⇣ = g ( X ) ⇣ = Band 2.2 ⇣ = ( ! ! X )( ! ! C ) Band 2.1 Band 2.1 Band 2.2 ⇣ = ⇣ ( ! ) ⇣ = ⇣ ( ! ) Figure 5: Bright solitons. Symmetric bright solitons for nonlinearity coefficient g < 0. (See thebullet points above in section 3.2 for a general explanation.) The minimal value ζ of ζ ( x ), indicatedby the solid dot in the leftmost diagram and given by (3.51), is at a simple root of Q ( ζ ) = Q ( ζ ; ω )when K = 0 so that ζQ ( ζ ) has a double root at ζ = 0. In the upper graphs (middle and right),the pair ( φ X ( x ) , φ C ( x )) travels from the open circle to one of the solid dots and back as x travelsfrom −∞ to ∞ . - - - - - - - - - - - - - - - ⇣ = !! X ! C X C X C X ( x ) C ( x ) X ( x ) C ( x ) x x ⇣ = g ( X ) Band 3.1 Band 3.2 ⇣ = ! ! X ⇣ = ( ! ! X )( ! ! C ) Band 3.2Band 3.1 ⇣ = ⇣ ( ! ) ⇣ = ⇣ ( ! ) Figure 6: Discontinuous solitons. Anti-symmetric solitons for which the exciton envelope φ X ( x )is discontinuous at its point of symmetry. (See the bullet points above in section 3.2 for a generalexplanation.) The far-field value ζ ∞ of ζ = gφ X , indicated by the open dots in the leftmost diagramand given by (2.20), is equal to a double root of the cubic Q ( ζ ) = Q ( ζ ; ω ) (see (2.23)) created bythe appropriate choice of constant K = K ( ω ). In the upper graphs (middle and right), as x travelsfrom −∞ to ∞ , the pair ( φ X ( x ) , φ C ( x )) travels along the cubic from an open circle to a solid dot,then jumps to the other solid dot, and then travels along the cubic to the other open circle. = ⇣ = ! ! X ! C ⇣ = g ( X ) ⇣ = ( ! ! X )( ! ! C ) ⇣ = ⇣ = ! ! X ⇣ = ⇣ = ⇣ = ⇣ ( ! ) ⇣ = ⇣ ( ! ) ⇣ = ⇣ ( ! ) ⇣ = ⇣ ( ! ) ! ! X ! C ⇣ = g ( X ) ⇣ = ( ! ! X )( ! ! C ) ⇣ = ⇣ = ! ! X ⇣ = ⇣ = ⇣ = ⇣ ( ! ) ⇣ = ⇣ ( ! ) ⇣ = ⇣ ( ! ) ⇣ = ⇣ ( ! ) Figure 7: Left: A superposition of the leftmost diagrams of Figures 4, 5, and 6, showing all bandssimultaneously in the positive detuning case, ω C > ω X . A given polariton system admits eitherbands 1.1, 1.2, and 3.2 if g > g < Right: In the negative detuningcase ω C < ω X , the band 3.2 of discontinuous solitons is absent. ! C ⇣ = ( ! ! X )( ! ! C ) dark g > g > g < g < g < ! ! ⇣ ! ! LP ! UP ! ! LP ! UP ! X ! X linear homogeneous polariton bandsnonlinear polariton soliton bandslower band upper band ! C ( ! C < ! X ) Figure 8: This is the analogous figure to Fig. 1 in the negative detuning case ω C < ω X . Note theabsence of band 3.2. Bands 1.1 and 3.2 merge to form a larger band of dark solitons for g > 0, which we callband D. This band was reported by the same authors in [15]. It consists of the interval15 ω , ω C ), where ω is defined by( ω − ω X )( ω − ω C ) = γ , ω < min { ω X , ω C } , (3.54)and coincides with the lower endpoint of a well-known band of homogeneous (constant in x )“lower polaritons” [18, Fig. 1] for the associated linear system obtained by setting g = 0and keeping all other parameters unchanged. The far-field amplitude of the soliton is givenby ζ ( x ) → ζ ∞ as | x | → ∞ , or gφ X → ω − ω X − γ ω − ω C as | x | → ∞ , (3.55)and ranges from 0 to ∞ as ω traverses the band ( ω , ω C ). In the case of negative detuning( ω C < ω X ), the discontinuous band 3.2 vanishes and the dark soliton is continuous on theentire band D. In the case of positive detuning ( ω X < ω C ), the exciton frequency ω X lieswithin band D and marks the transition from band 1.1 to band 3.2, where the soliton becomesdiscontinuous.Thus, in the positive-detuning case, the frequencies ω , ω X , and ω C have the followingsignificance for soliton band D: (cid:114) The value ω is the threshold frequency that marks the onset of a soliton. For frequenciesjust above this threshold (0 < ω − ω (cid:28) ω = ω . Thus, solitons at frequencies near the lower edge of the band are in the linearregime because the nonlinearity g | φ X | is negligible. (cid:114) The exciton frequency ω X is the transition frequency , at which the exciton field of thesoliton becomes discontinuous, as shown in Fig. 9. As ω exceeds ω X , the quantity (cid:36) X changes from negative to positive and the cubic relation between φ C and φ X gains twononzero roots at φ X = ± φ := ± (cid:112) (cid:36) X /g (Fig. 3, second row). The exciton field jumpsbetween these two roots exactly when the photon field vanishes, as shown in the rightmostgraphs of Fig. 9. (cid:114) The photon frequency ω C is the blowup frequency : as ω goes up to ω C , the far-fieldamplitude (3.55) tends to infinity.At the transition frequency ω X , the exciton field φ X ( x ) experiences an infinite slope whenits value equals zero, and the graph of | φ X ( x ) | has a cusp (Fig. 10). This soliton was foundnumerically in [21, Fig. 9]. Our equations (3.52–3.53) give an exact analytic expression ofthis soliton.In the negative-detuning case, the threshold and blowup frequencies persist, but thesoliton undergoes no transition to discontinuity.16 - - - - - X C C X - - - - - - - - C ( x ) C ( x ) X ( x ) X ( x ) ! < ! X ! > ! X ! Band 3.2 ! > ! X Band 1.1 ! < ! X ⇣ = ! ! X ! LP ! X ! C ⇣ = ⇣ ( ! ) | {z } Band D ⇣ = g ( X ) x x Figure 9: When ω C > ω X (positive detuning), band 1.1 of continuous dark solitons and band 3.2of discontinuous dark solitons merge to form the single band D of dark solitons. (See the bulletpoints in section 3.2 for a general explanation.) Their far-field value of gφ X = ζ ∞ is representedby the single expression Q ( ζ ∞ ) = 0 when the constant K in (2.23) is chosen so that Q has adouble root at ζ ∞ ; it is given explicitly by (3.55). The frequency ω is the threshold frequency,marking the onset of the soliton; the exciton frequency ω X marks the transition from continuousto discontinuous exciton field; and the far-field amplitude of the soliton blows up as ω goes up tothe photon frequency ω C . The values ± φ are the roots of φ C vs. φ X , and ± φ are the critical(local maximum and minimum) points. - - - C X X ( x ) C ( x ) ! = ! X | X ( x ) | x - - - x Figure 10: A dark soliton at the transition frequency ω = ω X . When ω C > ω X (positive detuning)the exciton frequency ω X lies within band D and marks where the exciton field φ X transitions fromcontinuous to discontinuous, as illustrated in Fig. 9. The soliton at ω = ω X is continuous, and thegraph of | φ C ( x ) | has a cusp at its nadir. The polariton system of equations (2.2–2.3), linearized about a soliton solution, has coeffi-cients that reflect the spatial dependence of the soliton wavefunction. Thus, an exact linearstability analysis cannot be based on the growth/decay of individual space-harmonic per-turbations. Utilizing the Laplace transform in time, we reduce the soliton linear stabilityto the invertibility of a two-component, time-independent Schr¨odinger operator in the inde-pendent variable x . The operator has a matrix potential M ( x, τ ), that carries the soliton17nformation and is required to be invertible for all nonreal values of the parameter τ = is ,where s is the Laplace independent variable. The time-symmetry of the problem, arisingfrom the losslessness, makes the classic stability requirement for (cid:61) τ > − ∂ xx + M ( x, τ ) for all nonreal τ . As | x | tends to infinity, thematrix potential M ( x, τ ) approaches exponentially an x -independent matrix M ( ∞ , τ ); inorder to check the invertibility of the Schr¨odinger operator, one has to show that it has nobounded null eigenfunctions. This is a challenging problem that is currently under study.We present a complete analysis of the linear stability of the soliton far fields in section 4.2. Inserting ψ X = ˜ ψ X e − iωt and ψ C = ˜ ψ C e − iωt into (2.2,2.3) yields i∂ t ˜ ψ X = (cid:0) − (cid:36) X + g | ψ X | (cid:1) ˜ ψ X + γ ˜ ψ C , (4.56) i∂ t ˜ ψ C = (cid:0) − (cid:36) C − ∂ xx (cid:1) ˜ ψ C + γ ˜ ψ X . (4.57)By taking ˜ ψ X and ˜ ψ C to be perturbed envelope functions˜ ψ X ( x, t ) = φ X ( x ) + ξ X ( x, t ) , (4.58)˜ ψ C ( x, t ) = φ C ( x ) + ξ C ( x, t ) , (4.59)one obtains equations for the perturbations ( ξ X , ξ C ), i∂ t ξ X = − (cid:36) X ξ X + gφ X (cid:0) ξ X + ¯ ξ X (cid:1) + γξ C + h.o.t. , (4.60) i∂ t ξ C = (cid:0) − (cid:36) C − ∂ xx (cid:1) ξ C + γξ X , (4.61)in which the omitted terms are higher than linear order in ξ X and ¯ ξ X . By taking the Laplacetransform of these equations and their conjugates, one obtains the system (cid:0) is + (cid:36) X − gφ X (cid:1) ˆ ξ X − gφ X ˆ¯ ξ X − γ ˆ ξ C = 0 (4.62) (cid:0) − is + (cid:36) X − gφ X (cid:1) ˆ¯ ξ X − gφ X ˆ ξ X − γ ˆ¯ ξ C = 0 (4.63) (cid:0) is + (cid:36) C + ∂ xx (cid:1) ˆ ξ C − γ ˆ ξ X = 0 (4.64) (cid:0) − is + (cid:36) C + ∂ xx (cid:1) ˆ¯ ξ C − γ ˆ¯ ξ X = 0 (4.65)or, in matrix form, gφ X − (cid:36) X − is gφ X γ gφ X gφ X − (cid:36) X + is γγ − ∂ xx − (cid:36) C − is γ − ∂ xx − (cid:36) C + is ˆ ξ X ˆ¯ ξ X ˆ ξ C ˆ¯ ξ C = 0 . (4.66)18otice that the field ( φ X , φ C ) occurs in the matrix only through ζ ( x ) = gφ X ( x ) .We write the inhomogeneous version of the linearized problem (4.66) as (cid:34) A γγ B (cid:35) (cid:34) (cid:126)ξ X (cid:126)ξ C (cid:35) = (cid:34) (cid:126)f (cid:126)f (cid:35) , (4.67)in which (cid:104) (cid:126)ξ X , (cid:126)ξ C (cid:105) T = (cid:104) ˆ ξ X , ˆ¯ ξ X , ˆ ξ C , ˆ¯ ξ C (cid:105) T (4.68)and A and B are the matrix operators A = (cid:20) α − τ ζζ α + τ (cid:21) , B = (cid:20) − ∂ xx − (cid:36) C − τ − ∂ xx − (cid:36) C + τ (cid:21) , (4.69)where α := 2 gφ X − (cid:36) X = 2 ζ − (cid:36) X (4.70)and τ = is is a complex frequency. Equation (4.67) can be solved for the photon component: (cid:0) B − γ A − (cid:1) (cid:126)ξ C = (cid:126)f − γA − (cid:126)f . (4.71)The operator on the left is a vector Schr¨odinger operator, B − γ A − = − ∂ xx − (cid:20) (cid:36) C + τ (cid:36) C − τ (cid:21) − γ α − ζ − τ (cid:20) α + τ − ζ − ζ α − τ (cid:21) . (4.72)Thus, equation (4.71) becomes − ( ∂ xx + M ( x, τ ) (cid:126)ξ C = (cid:126)f − γA − (cid:126)f , (4.73)where M ( x, τ ) = (cid:20) (cid:36) C + τ (cid:36) C − τ (cid:21) + γ α ( x ) − ζ ( x ) − τ (cid:20) α ( x ) + τ − ζ ( x ) − ζ ( x ) α ( x ) − τ (cid:21) . (4.74)In the definition of M ( x, τ ), both ζ ( x ) and α ( x ) = 2 ζ ( x ) − (cid:36) X are functions of x thatexponentially converge to limiting values as x → ±∞ . Thus the matrix potential M ( x, τ ) isan exponentially localized perturbation of its large- | x | value M ( ∞ , τ ). Linear stability of a solution ( ψ C ( x ) , ψ X ( x )) , corresponding to ζ ( x ) is obtained if,for all τ in the upper half-plane, the vector Schr¨odinger operator ∂ xx + M ( x, τ ) admits noextended states and no bound states , that is, bounded vector functions (cid:126)ξ C ( x ) satisfying theequation ( ∂ xx + M ( x, τ )) (cid:126)ξ C = 0 . (4.75)19 .2 Linear stability of the soliton far fields Soliton far-field solutions , or simply soliton far fields, are homogeneous ( x -independent andtime-harmonic) solutions of the polariton system of equations (2.2–2.3), that are asymptoticto soliton solutions of the system as x tends to ±∞ . They have form ( φ ∗ X , φ ∗ C ) e − iωt , where ω isthe frequency and the pair ( φ ∗ X , φ ∗ C ) is an equilibrium point of the ODE system (2.11)–(2.12).We have seen that the value ζ ∗ of the variable ζ = gφ X of a soliton far field is independentof whether x → ∞ or x → −∞ . The value ζ ∗ is a double root of the quartic polynomial ζQ ( ζ ) in the ODE (2.21). This root takes on one of two values for any given soliton. Onevalue, ζ ∗ = 0 (simple root of Q ( ζ ) when K = 0) is taken by the solitons of frequency bands2.1 and 2.2. The other value, ζ ∗ = ζ ∞ (double root of Q ( ζ )) is taken over the frequencybands 1.1, 1.2, 3.1 and 3.2. Nonzero simple roots of the cubic Q ( ζ ) do not correspond tosoliton far-field values, and they do not satisfy the original system (2.11, 2.12). They aregenerated from the derivation of the ODE (2.21) for ζ , which involves multiplying (2.11) and(2.12) by the derivatives φ (cid:48) X and φ (cid:48) C , which vanish when φ X and φ C are constant.For soliton far fields, the Schr¨odinger equation (4.75) reduces to the constant-coefficientproblem ( ∂ xx + M ( ∞ , τ )) (cid:126)ξ C ( x ) = 0 , (4.76)by putting ζ ( x ) = ζ ∗ and α ( x ) = α ∗ . The far-field dispersion relation , relating wave number k and frequency τ as | x | → ∞ for the linearization (4.76), is obtained by replacing ∂ xx with − k and setting the determinant to zero, D ( k, τ ) = det (cid:0) k I − M ( ∞ , τ ) (cid:1) = τ − (cid:0) α ∗ + β + 2 γ − ζ ∗ (cid:1) τ + (cid:0) γ − α ∗ βγ + β ( α ∗ − ζ ∗ ) (cid:1) = 0 (4.77)in which β = k − (cid:36) C and α ∗ = 2 ζ ∗ − (cid:36) X . D ( k, τ ) is a function of k and τ .The stability condition for homogeneous solutions obtained in the previous section canbe rephrased as follows: For all τ ∈ C \ R , the matrix M ( ∞ , τ ) has no eigenvalues k in [0 , ∞ ) , or, equivalently, D ( k, τ ) (cid:54) = 0 for such τ and k . The proof of the following theorem is given in section 6. Theorem 3. Let ( ψ X ( x, t ) , ψ C ( x, t )) = ( φ X ( x ) , φ C ( x )) e − iωt be a solution of the nonlinearpolariton system (2.2,2.3) such that ( φ X ( x ) , φ C ( x )) has limits as x → ±∞ . All such solutionsare described in Theorem 2, and one has ( φ X ( x ) , φ C ( x )) → ± ( φ ∗ X , φ ∗ C ) as x → ±∞ . The function pair ( φ ∗ X , φ ∗ C ) e − iωt is a homogeneous solution to (2.2,2.3) that is linearlystable if ω ∈ band 1.1, 2.1, 2.2, or 3.2,unstable if ω ∈ band 1.2 or 3.1.In bands 2.1 and 2.2, ( φ ∗ X , φ ∗ C ) = (0 , , and in the other bands, ( φ ∗ X , φ ∗ C ) = ± ( φ ∞ X , φ ∞ C ) ,defined in (2.13). emark: The determinant (4.77) can be obtained directly (as a function of s as opposedto the above τ = − s ) as the determinant of the matrix (4.66), in which ∂ xx is replaced by − k . The stability condition, rephrased for the s variable is: Linear stability at all modes k requires that the two roots of the determinant D , considered as a quadratic in the variable s , be negative or zero for all real values of k , i.e. for all values of β that satisfy β ≥ − (cid:36) C . We use this approach in the proof in section 6. This section contains the proofs of the six classes of solitons described in Theorem 2 ofsection 3.The derivation of these solitons is simplified by passing to a normalized frequency variable η that conveniently parameterizes the operating frequency ω , η = γ ( ω − ω X )( ω − ω C ) = (cid:36) X (cid:36) C γ . (5.78)This expression, which is quadratic in ω , produces generically two frequencies for the samevalue of η , a first indication of the fact that exciton-polariton soliton solutions typically comein pairs. One only needs to consider values η ≥ η min = ( ω X − ω C ) γ , (5.79)that are at or above the minimum η min of the quadratic and thus produce real frequencies.The non-dimensional frequency (cid:36) X is a natural scaling factor for ζ : ζ = (cid:36) X u , ζ ∞ = (cid:36) X u ∞ , u ∞ = 1 − η . (5.80)Using u instead of ζ greatly simplifies the algebraic computations of solitons. In the newvariables, the ODE (2.21) for ζ becomes an ODE for u ,(3 u − u (cid:48) = − (cid:36) C u (cid:104) u − (cid:16) − η (cid:17) u + (cid:16) − η (cid:17) u + ˜ K (cid:105) , = − (cid:36) C u B ( u ) when (cid:36) X (cid:54) = 0 . (5.81)The polynomial B ( u ) is related to Q ( ζ ) (see (2.23)) by B ( u ) = − (cid:36) C (cid:36) X Q ( (cid:36) X u ) . (5.82)The phase space of the ODE (5.81) is one-dimensional (the u -axis). Because u (cid:48) appearssquared, a soliton solution of the ODE is obtained through standard phaseline analysis by connecting a double root of the quartic polynomial uB ( u ) on the right side of the ODE with asimple root of uB ( u ) . A solution connecting these consecutive roots is possible provided thatthe interval between the two roots does not contain the singular value u = 1 / 3; we call this21he connectivity condition . In addition, the sign condition requires the positivity ofthe right side of the equation over the interval between the two roots; it can be expressed as − (cid:36) C ( uB ( u )) (cid:48)(cid:48) | u =double root ≥ . (sign condition) (5.83)The solution u ( x ) approaches the double root exponentially slowly as x approaches ±∞ ;thus the double root signifies the far-field amplitude of the soliton and is thus equal to u ∞ .The simple root u is the extremal value (maximum or minimum) of u ( x ) and is attained ata finite value x ∗ . The solution u ( x ) is symmetric about x ∗ , and has local quadratic behaviorthere. Because of the invariance of the polariton system under a shift x (cid:55)→ x − x ∗ , we willhenceforth take x ∗ = 0.Since uB ( u ) vanishes at u = 0, it is guaranteed that the interval between two consecutiveroots is either positive or negative, so that the solution u ( x ) is of one sign. This allows oneto choose the sign of g appropriately so that ( (cid:36) X /g ) u > φ X = ± (cid:112) ( (cid:36) X /g ) u . If thesimple root u is nonzero, then the exciton envelope field is symmetric and of one sign, φ X ( x ) = ± (cid:114) (cid:36) X g u ( x ) , if u (cid:54) = 0 . If the simple root u is equal to zero, then the exciton envelope field is anti-symmetric, φ X ( x ) = ± sgn( x ) (cid:114) (cid:36) X g u ( x ) , if u = 0 . 1. Dark solitons. These are solutions that connect a double nonzero root of uB ( u ),(far-field value) with the zero root (nadir).The following factorization is key to the analysis; B ( u ) = ( u − u ∞ ) ( u − u ) , (cid:40) u ∞ = 1 − η u = η , ˜ K = − u ∞ u . (5.84)One observes that the singular value u = 1 / u ∞ and the nonzero simple root u and thus obstructs a soliton connection between them: u ∞ = + 2 (cid:16) − η (cid:17) ,u = − (cid:16) − η (cid:17) . (5.85)In order for u ∞ to be connectible to the other root u = 0 of uB ( u ), the following two mutuallyequivalent conditions must hold: u ∞ < , i.e., 0 < η < . (connectivity condition) (5.86)Furthermore, one obtains from (5.83), the sign condition − (cid:36) C u ∞ ( u ∞ − ) > . (sign condition) (5.87)22s a result of the positivity of η , the frequencies (cid:36) X and (cid:36) C must have the same sign. Thereare therefore two cases ( (cid:36) X < , (cid:36) C < 0) and ( (cid:36) X > , (cid:36) C > (cid:36) X and (cid:36) C are both negative, or ω < min { ω X , ω C } , conditions (5.86,5.87) necessitate u ∞ < 0, which by u ∞ = 1 − η − is equivalent to 0 < η < 1. This yields theband { < η < , ω < min { ω X , ω C }} . (band 1.1) (5.88)Thus for each η between 0 and 1 the ODE (5.81) has a soliton solution u ( x ) with nadir equalto the simple root u = 0 and far-field value equal to the double root u ∞ .When converting η back to the variable ω through η = (cid:36) X (cid:36) C /γ , the condition ω < min { ω X , ω C } determines the choice of frequency ω as the lower of the two solutions of ηγ = ( ω − ω X )( ω − ω C ). This results in band 1.1 stated in Theorem 2 (see 3.46) anddepicted in Fig. 4. The interval 0 < η < < (cid:36) X (cid:36) C < γ in (3.46).In converting u back to the variable ζ , notice that u ∞ and (cid:36) X are both negative so that ζ ∞ = (cid:36) X u ∞ > 0, and thus ζ ( x ) = (cid:36) X u ( x ) > x since u is of one sign on theinterval ( u ∞ , g is determined by 0 < ζ = gφ X , so g > ζ ∞ .In the case that (cid:36) X and (cid:36) C are both positive, or ω > max { ω X , ω C } , conditions (5.86,5.87) necessitate 0 < u ∞ < , which by u ∞ = 1 − η − is equivalent to 1 < η < . This yieldsthe band { < η < , ω > max { ω X , ω C }} . (band 1.2) (5.89)This results in band 1.2 stated in Theorem 2 (see 3.46) and depicted in Fig. 4. The interval1 < η < corresponds to the condition γ < (cid:36) X (cid:36) C < γ in (3.46). Since again ζ ∞ = (cid:36) X u ∞ > 0, one has gφ X ( x ) = ζ ( x ) > x , and again g > 2. Bright solitons. These are solutions that connect zero as a double root of uB ( u )(far-field value) to a simple root (peak).Assuming uB ( u ) has a double root at 0, (5.81) takes the form(3 u − u (cid:48) = − (cid:36) C u (cid:104) u − (cid:16) − η (cid:17) u + 1 − η (cid:105)(cid:124) (cid:123)(cid:122) (cid:125) Quadratic A ( u ) , when (cid:36) X (cid:54) = 0 . (5.90)We are interested in the ranges of η for which roots of the quadratic A ( u ) are to the left of thesingular point u = and thus can be connected with the double root at u = 0 (connectivitycondition). By a simple argument, there is either one root u or no root in the half-line u < . The range of η for which this root is present is0 < η < , (cid:40) < u < , when 1 < η < ,u < , when 0 < η < . (5.91) Write ( u − = − η ( u − ) and examine how the line cuts the quadratic, as η is varied. , so one computes u = 1 − η (cid:104) + (cid:113) − η (cid:105) . (5.92)As η decreases from η = to η = 1, the root u descends from u = to u = 0, then turningnegative as η decreases from the value 1. In the limit η → +0, u → −∞ .The condition η > (cid:36) X (cid:36) C > 0, so that, as before, either ( (cid:36) X < , (cid:36) C < (cid:36) X > , (cid:36) C > u = 0, (cid:36) C (1 − η ) < . (sign condition) (5.93)Putting these requirements together, we obtain two soliton bands, (cid:40) band 2.1 = { < η < , (cid:36) X < , (cid:36) C < } u > , ζ = (cid:36) X u < , g < , (5.94) (cid:40) band 2.2 = { < η < , (cid:36) X > , (cid:36) C > } u < , ζ = (cid:36) X u < , g < . (5.95)As in the previous case of dark solitons, the endpoints of the frequency bands are imposedby the bounds of η and the relation ηγ = (cid:36) X (cid:36) C , and the sign of g coincides with the signof ζ , which is negative in these cases.The minimal (negative) value ζ of ζ ( x ) = (cid:36) X u ( x ) is equal to (cid:36) X u , which from (5.92)is equal to ζ := (cid:36) X u = (cid:36) X − γ (cid:36) C (cid:104) + (cid:113) − (cid:36) X (cid:36) C γ (cid:105) , (5.96)and since g < 0, one obtains the peak value of | g | φ X stated in the theorem. 3. Discontinuous solitons. In the derivation of the dark solitons, the case u ∞ > wasexcluded because the connectivity of u ∞ to the zero root was broken by the singularity at u = . Consider instead the u -interval between u ∞ and u = 1, which does not contain 1 / u ∞ ) of B ( u ). The corresponding ζ -interval connects ζ ∞ an (cid:36) X and doesnot contain (cid:36) X / ζ ∞ ) of ζQ ( ζ ).This ζ -interval corresponds to two φ X -intervals, connecting ± φ ∞ X with ± φ := ± (cid:112) (cid:36) X /g and not containing ± (cid:112) (cid:36) X / (3 g ) , where φ ∞ X > g ( φ ∞ X ) = ζ ∞ . Naturally, g must take the sign of ζ ∞ , and thus the cubic (2.22) giving φ C as a function of φ X vanisheswhen φ X = ± φ .Given that the sign condition holds, a discontinuous soliton is constructed by taking asolution of (2.21) for which ζ ( x ) travels from φ to φ ∞ X as x travels from 0 to ∞ , then setting φ X ( x ) = (cid:112) ζ ( x ) /g for x > φ X ( x ) = − (cid:112) ζ ( − x ) /g for x < φ C ( x ) = γ − φ X (cid:0) (cid:36) X − gφ X (cid:1) for x ∈ R . φ X ( x ) , φ C ( x )) is antisymmetric about x = 0 and satisfies the pair(2.11,2.12) for x (cid:54) = 0. Since φ C vanishes when φ X = ± φ , setting φ C (0) = 0 makes thefield φ C ( x ) continuous; this together with antisymmetry makes φ C ( x ) continuously differen-tiable at x = 0. Thus (2.12) is satisfied in the sense of distributions, even through x = 0,and the jump of φ (cid:48)(cid:48) C ( x ) across x = 0 is computed from the ODE:[ φ (cid:48)(cid:48) C ( x )] x =0 = 2 γ [ φ X ( x )] x =0 = 4 γ (cid:112) (cid:36) X / . (5.97)Violation of the connectivity condition means u ∞ > , i.e., η > or η < . (no-connectivity condition) (5.98)The sign condition (5.87) still applies and reduces to (cid:36) C < , (5.99)as a result of u ∞ > . The sign of (cid:36) X is opposite to the sign of η , as follows from thedefinition of η (5.78). From the relation u ∞ = 1 − η − , we obtain two frequency bands, onefor η > / 2, and one for η < 0. For η > / (cid:40) band 3.1 = { η > , ω < min { ω X , ω C }} u ∞ > , ζ ∞ = (cid:36) X u ∞ < , g < . (5.100)In this band, the far-field value of u is u ∞ = 1 − η − , so the range of u ( x ) is ( u ∞ , 1) andthus ζ ( x ) = (cid:36) X u ( x ) has a far-field value of (cid:36) X − γ /(cid:36) C and, since (cid:36) X < 0, its range isequal to negative interval ( (cid:36) X , ζ ∞ ). The corresponding discontinuous soliton is bright since | ζ ∞ | < | (cid:36) X | .In the case η < 0, one obtains (cid:40) band 3.2 = { η < , ω X < ω < ω C } u ∞ > , ζ ∞ = (cid:36) X u ∞ > , g > . (5.101)The far-field amplitude of u is again u ∞ = 1 − η − , so the range of u ( x ) is (1 , u ∞ ) Since (cid:36) X > 0, the range of ζ ( x ) is the positive interval ( (cid:36) X , ζ ∞ ). This section is devoted to a proof of Theorem 3, using the notation introduced there. Thedeterminant D ( ∞ , τ ) (4.77) with τ = is is D := s + ( α + β + 2 γ − ζ ) s + γ − αβγ + α β − ζ β , (6.102)with ζ = 0 or ζ = ζ ∞ . Linear stability at all modes k requires that the two roots of thedeterminant D , considered as a quadratic in the variable s , be negative or zero for all realvalues of k , i.e. for all values of β that satisfy β ≥ − (cid:36) C . This is equivalent to the followingthree conditions: 25. The product of the roots is positive or zero γ − αβγ + α β − ζ β ≥ , for all β ≥ − (cid:36) C . (6.103)2. Their sum of the roots is negative or zero α + β + 2 γ − ζ ≥ , for all β ≥ − (cid:36) C . (6.104)3. The discriminant is positive or zero( α + β + 2 γ − ζ ) − γ − αβγ + α β − ζ β ) ≥ , for all β ≥ − (cid:36) C . (6.105)Inequality (6.105) is the hardest of the three conditions to analyze. Through algebraicmanipulation, it is recast as( α − β − ζ ) + 4 γ ( α + β + 2 αβ − ζ ) ≥ , for all β ≥ − (cid:36) C . (6.106)The three inequalities together constitute necessary and sufficient conditions for theasymptotic values ζ = 0 or ζ = ζ ∞ of a soliton solution to be a linearly stable homoge-neous solution. We refer to these inequalities below as the first, second, and third stabilityconditions. ζ = 0 . The left side of each of the three inequalities above is either a perfect square or a sum ofsquares (see the third inequality in its recast form (6.106)). Thus, they are all satisfied, andso the soliton far-field solutions for bands 2.1 and 2.2 are stable. ζ = ζ ∞ . First stability condition. For ζ = ζ ∞ , the left side of the inequality (6.103) factors to (cid:0) γ − ( α ∞ + ζ ∞ ) β (cid:1) (cid:0) γ − ( α ∞ − ζ ∞ ) β (cid:1) ≥ , (6.107)in which α ∞ is the value of α at ζ = ζ ∞ . The definition of η gives directly (cid:36) X = ηγ (cid:36) C , (6.108)from which one obtains easily α ∞ + ζ ∞ = γ (cid:36) C (2 η − , α ∞ − ζ ∞ = − γ (cid:36) C . (6.109)Inserting these into the above inequality and recalling that β = k − (cid:36) C , yields (cid:104)(cid:0) η − (cid:1) k (cid:36) C + η − (cid:105) k (cid:36) C ≥ k . (6.110)26he sign distribution of the left of the inequality reveals that the inequality is satisfied inexactly two regimes (cid:40) η min ≤ η ≤ , (cid:36) C < ≤ η ≤ , (cid:36) C > . (6.111)The homogeneous solutions corresponding to the far-field values of the solitons in bands1.1 and 3.2 are in the first regime. Those corresponding to band 1.2 are in the second regime.Thus, the far-field values of all these dark solitons pass the first test for linear stability. Onthe other hand, the homogeneous solutions corresponding to the far-field values of the solitonsin band 3.1 are outside these two regimes and therefore are not linearly stable.The first stability condition (6.103) poses a simple restriction on the homogeneous solu-tions ζ = ζ ∞ , as one observes that it is a quadratic inequality in the variable β/γ ,( α − ζ ) (cid:32) βγ (cid:33) − α (cid:32) βγ (cid:33) + 1 ≥ , for all β ≥ − (cid:36) C . (6.112)Necessarily, α − ζ ≥ . (6.113)This is an interesting inequality. Factoring and recalling that α = 2 ζ − (cid:36) X , it becomes,( ζ − (cid:36) X )( ζ − (cid:36) X ) ≥ ζ = ζ ∞ that lie outside the open intervalbetween (cid:36) X and (cid:36) X / 3. To the right of this interval α > 0, while α < ζ is tothe left of the interval. Second stability condition. Inequality in (6.104) follows immediately from the obtained requirement of the first stabilitycondition, α − ζ ≥ Third stability condition. The far-field solutions for bands 1.1 and 3.2 satisfy the first two stability conditions. Weshow now that they also satisfy the third condition. It suffices to show that the second termin parentheses (call it A ) in (6.106) is positive or zero. The proof is based on the fact thatboth solutions have (cid:36) C < A = ( α + β ) − ζ = ( α + β + ζ )( α + β − ζ ) . (6.115)Inserting the expression (6.110) for α ± β , we obtain A = ( γ (cid:36) C (2 η − 3) + k − (cid:36) C )( − γ (cid:36) C + k − (cid:36) C ) . (6.116)27he term 2 η − (cid:36) C < 0, every term in each of the twoparenthesis is positive or zero.The expression for A above is quadratic in k with roots − γ (cid:36) C (2 η − 3) + (cid:36) C and γ (cid:36) C + (cid:36) C .For the far-field solutions for band 1.2, necessarily (cid:36) C > 0, and thus both roots are positive.Giving k a value between these roots makes the quadratic expression negative. The thirdstability condition is thus violated, so the far-field solution for band 1.2 is unstable. Table 1gives a summary of linear far-field stability of all solitons.Table 1: Soliton propertiesBand Bright/Dark Linearly stable far field η domain3.1 Neither no < η < ∞ < η < < η < η min < η < < η < < η < We have studied soliton solutions in a polariton condensate and have derived the completespectrum of static one-dimensional solitons. The stationarity property (harmonic with anon-traveling envelope) permits a reduction of the polariton equations to a real first-orderordinary differential equation. This allows symbolic integration of the polariton equations,resulting in exact analytical formulae for stationary polariton solitons. For attractive exciton-exciton interactions we find two bands of bright solitons while for repulsive interactions wefind two bands of dark solitons. In addition, a band of dark solitons with a discontinuousexciton field at the soliton center (discontinuous solitons) is found for attractive interactionsand a band of discontinuous bright solitons with a nonzero background field is found forattractive interactions. One-dimensional solitons have been shown to be realizable in apolariton waveguide through detuning of the microcavity in the x (transverse) direction [8].The system of two equations for the exciton and photon wavefunctions can, in general, notbe reduced to the Gross-Pitaevskii model for a single wavefunction representing polaritons.A reduction is possible in certain regimes, and has been given in [15, section III], where it isshown to apply at the left end of band 1.1 of dark solitons for g > ζ ≈ γ ). Modeling the full range of solitons requires the system of two equations.Specifically, for the bands of discontinuous solitons, one field vanishes where the other oneis nonzero, and this phenomenon obviously lies outside the parameter regime of validity ofthe Gross-Pitaevskii model.The six bands of solitons we discover are the static members of presumably much largerclasses of solutions of the one-dimensional polariton system that include traveling and forcedsolitons. For example, one-dimensional stable traveling bright solitons for g > 0, sustainedby an optical source, are reported in [8]. Finding traveling soliton solutions of the polariton28quations, even in one spatial dimension, is not a simple matter. This is because the excitonand polariton wavefunctions in (2.8) are in general complex, resulting in a fourth-ordersystem of real ODEs.Two-dimensional polaritons exhibit an abundance of interesting phenomena that promisesome mathematical challenges. Unlike the nonlinear Schr¨odinger equation, the polaritonsystem (1.1) with κ X = κ C = 0 is not invariant under the Galilean transformation( ψ X ( x , t ) , ψ C ( x , t )) (cid:55)→ ( ψ X ( x − ξ t, t ) , ψ C ( x − ξ t, t )) e i ( k · x − ωt ) , (7.117)in which ξ = (cid:126) m C k and ω = (cid:126) m C | k | . The upper left entry of the matrix in (1.1) gains atransport term − i ξ · ∇ ψ X and a frequency shift ω X (cid:55)→ ω X − ω . The scaling transformation( ψ X ( x , t ) , ψ C ( x , t )) (cid:55)→ λ ( ψ X ( λ t, λ x ) , ψ C ( λ t, λ x )) , (7.118)which preserves the cubic nonlinear Schr¨odinger equation, effects a transformation of thepolariton equations through a scaling of the frequency parameters,( ω X , ω C , γ ) (cid:55)→ λ ( ω X , ω C , γ ) . (7.119)The result is a simple scaling by λ of both the ω and the ζ axes in the depiction of the bandstructure of solitons in Fig. 1. Acknowledgment. This work was partially supported by the European Union’s FP7-REGPOT-2009-1 project “Archimedes Center for Modeling, Analysis and Computation”(grant agreement n. 245749) and by the US National Science Foundation under grants NSFDMS-0707488 and NSF DMS-1211638. We acknowledge discussions on the physics andexperimental aspects of polariton condensates with P. Savvidis, G. Christmann, F. Marchetti. References [1] T Ackemann, W. J. Firth, and G.-L. Oppo. 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