A method is proposed to solve the challenging problem of determining the
supratransmission threshold (onset of instability of harmonic boundary driving
inside a band gap) in multicomponent nonintegrable nonlinear systems. It is
successfully applied to the degenerate three-wave resonant interaction in a
birefringent quadratic medium where the process generates spatial gap solitons.
No analytic expression is known for this model showing the broad applicability
of the method to nonlinear systems.Comment: 4 pages, 3 figure

Anghel-Vasilescu, P.

Dorignac, J.

Geniet, F.

Leon, J.

Taki, M.

English

arXiv

http://arxiv.org/abs/1008.1920International audienceA method is proposed to solve the challenging problem of determining the supratransmission threshold (onset of insability of harmonic boundary driving inside a band gap) in multicomponent non-integrable nonlinear systems. It is successfully applied to the degenerate three-wave resonant interaction in a birefrigent quadratic medium where the process generates spatial gap solitons. No analytic expression is known for such nonlinear solitary waves showing the broad applicability of the method to nonlinear systems

Anghel-Vasilescu, Petrutza

Dorignac, Jerome

Geniet, Frédéric

Leon, Jerome

Taki, Majid

HAL-IN2P3

arXiv:1008.1920v1  [nlin.PS]  11 Aug 2010Nonlinear supratransmission in multicomponent systemsP. Anghel-Vasilescu, J. Dorignac, F. Geniet, J. Leon, M. Taki∗Laboratoire de Physique The´orique et AstroparticulesCNRS-IN2P3-UMR5207, Universite´ Montpellier 2, 34095 Montpellier (France)A method is proposed to solve the challenging problem of determining the supratransmissionthreshold (onset of instability of harmonic boundary driving inside a band gap) in multicomponentnonintegrable nonlinear systems. It is successfully applied to the degenerate three-wave resonantinteraction in a birefringent quadratic medium where the process generates spatial gap solitons.No analytic expression is known for this model showing the broad applicability of the method tononlinear systems.PACS numbers: O5.45.Yv, 42.65.Tg Phys. Rev. Lett. 105 (2010) 074101Introduction. Nonlinear supratransmission (NST) ina medium possessing a natural forbidden band gap is aprocess by which nonlinear structures, gap solitons, aregenerated by an applied periodic boundary condition ata frequency in the band gap. Discovered in the pendulachain (sine-Gordon model) [1], and further studied forfully discrete chain in Refs. [2] and [3], it has been ap-plied, among others, in Bragg media (coupled mode equa-tions in Kerr regime) [4] allowing to explain the experi-ments of Ref. [5], and also to coupled-wave-guide arrays(nonlinear Schro¨dinger model) [6][7]. Nonlinear supra-transmission results from an instability of the evanescentwave profile created by the driving [8][9] that manifestsitself above a threshold amplitude. Today this thresholdhas been obtained in single component systems by mak-ing use of the explicit solution of the model equation andseeking its maximum allowed amplitude at the boundary.Predicting the threshold value is of fundamental im-portance for physical applications such as soliton gener-ation or conception of ultrasensitive detectors. Indeed,on the one side NST is a very efficient means to generategap solitons: while an incident single pulse with carrierwave at forbidden frequency would be mainly reflected,an incident continuous wave excitation easily producesgap soliton as experimentally shown in Bragg media [5].On the other side such systems seeded by a CW excita-tion slightly below the threshold will be extremely sensi-tive to any applied signal, detected either through gen-eration of gap solitons or by bistable behavior, see e.g.[10].We address in this Letter the practical question ofevaluating NST thresholds in multicomponent systems(where the instability of either wave separately caninduce soliton formation in all channels), and more-over when the system has no explicit solution allow-ing for threshold prediction. This is the case withsecond harmonic generation in a birefringent mediumwith quadratic nonlinearity [11]. The model is a two-component system which does not possess analytic ex-pression of solitonlike solutions and which is a key modelin order to study their existence and stability; see, e.g.,[12].We shall develop a method based on an asymptoticsolution obtained by asymptotic series expansion, whichprovides an accurate NST threshold prediction. As NSTrequires driving in the forbidden band, the linear evanes-cent wave is the natural keystone upon which to build theseries. The method is restricted to neither the specificcase of second harmonic generation nor to the quadraticnature of the nonlinearity. Moreover, it can be applied toa wide class of nonintegrable multicomponent nonlinearsystems since it does not require known analytical ex-pressions for their solutions. The method thus furnishesa practical tool highly interesting for further applicationsin any multi-component coupled-wave system.0 5 10 1512|φ||ψ|xz z5060700 10 20 0 10 20xgenerationgap solitonevanescentwave ABFIG. 1. (color online) Left: intensity plots of a typical numer-ical simulation of the D3W system (1) with α = 1.1 subjectto boundary conditions (2) with A = 5, B = 1.3 [the crosson the (A,B) plot]. The maximum amplitudes of the emit-ted soliton are |ψ|m ∼ 4.5 and |φ|m ∼ 5.5. Right: thresholdcurve in the (A,B) plane where dots result from numericalsimulations [13] and the curve is given by the solution of (6).Birefringent gap solitons. Let us consider a birefrin-gent medium in permanent regime, namely assuming per-fect frequency matching. In that case, degenerate spatialthree-wave model (D3W) reduces to [11][12]i∂zψ +α2∂2xψ − δψ + φ2 = 0,i∂zφ+ ∂2xφ+ ψφ∗ = 0. (1)where φ(x, z) [respectively ψ(x, z)eiδz ] is the scaled staticenvelope of the signal wave with frequency ω and wave2number k [respectively second harmonic at frequencyω′ = 2ω and wave number k′] and δ is the missmatchwavenumber in the propagation direction z defined byk′ = 2k − δ. Last, x is the transverse direction andα = 1 + δ/k′ by definition. The system (1) is subject tothe boundary conditionψ(0, z) = Ae2iz , φ(0, z) = Beiz (2)on the strip x > 0, z ∈ [0, L], with vanishing conditionsas x → ∞ and now with normalization k′ = 2 (that isα = 1 + δ/2) [14].For the sake of simplicity, we only consider here theboundary conditions (2), although the wave number mayin general be chosen different from 1. We see on Fig.1the generation, through the evanescent coupling, of bire-fringent gap solitons (BGS) formation and propagationabove a threshold curve in the amplitude plane (A,B). Itis worth pointing out that the present situation is funda-mentally different from studies of initial-value problemswhere field values are given at z = 0, as, e.g., in the the-oretical prediction [15] and experimental realization [16]for nondegenerate 3-wave interaction.Asymptotic series solution. Given the boundary val-ues (2), we seek stationary solutions of the formψ(x, z) = u(x)e2iz , φ(x, z) =√α v(x)eiz , (3)with real-valued functions u and v vanishing at infinity(the factor√α has been included for convenience). Thesystem (1), now with δ = 2(α− 1), provides for u and vthe following parameter free equations∂2xu− 4u+ 2v2 = 0, ∂2xv − v + uv = 0, (4)Treating nonlinear terms as perturbative, we first solvethe linearized equation for v and obtain v = βe−x. Sub-stituting this result in the equation for u, we find that thev2 term is resonant and generates a solution of the formu = (µ + β2x/2)e−2x where µ and β are two arbitraryconstants. This is the general solution of the quasilinearsystem ∂2xu − 4u + 2v2 = 0, ∂2xv = v, that vanishes asx→∞.The structure of (4) now indicates that u and v maybe expressed as the following asymptotic seriesu =∞∑n=0Pn(x)e−2(n+1)x, v =∞∑n=0Qn(x)e−(2n+1)x,Q0(x) = β, P0(x) = µ+ β2x/2. (5)By inspection, the polynomials Pn(x) and Qn(x) are ofdegree n (except P0 of degree 1) and obey a systemof differential-recurrence equations obtained by replac-ing (5) in (4). Their coefficients are then recursivelygiven in terms of the two independent parameters β andµ. To ensure an accurate determination of the thresh-old curve, we have evaluated the first 17 terms of theseries (5), which takes up to a minute on a PC com-puter (with MAPLE or MATHEMATICA). The firstones are e.g. Q1 = −3β3/64 − µβ/8 − β3x/16 andP1 = β2(24µ + 17β2)/576 + xβ4/48. The advantage ofthis method is to be applicable to any system driven ina forbidden band, which is essential when no soliton so-lution is known. Notice that, in the absence of resonantterms in the equations, the polynomials involved in theasymptotic series would simply be constants as e.g. inthe Manakov system below.NST threshold prediction. Once the series (5) hasbeen determined up to a given truncation order N , im-posing the boundary conditions u(0) = A and v(0) = Bleads to the two driving amplitudes A(β, µ) and B(β, µ)explicitly expressed in terms of β and µ. Assuming thatthe supratransmission threshold curve is given by themaximum value of one of those, the other one beingheld constant, we can use a Lagrange parameter λ andwrite the extremum condition as ∂µA − λ∂µB = 0 and∂βA − λ∂βB = 0. This finally leads to the condition ofvanishing JacobianJ(β, µ) = (∂βA)(∂µB)− (∂µA)(∂βB) = 0. (6)With a high enough truncation order (N = 16 here), thethreshold curve is best obtained as the zero contour ofthe surface J(β, µ) plotted parametrically as a functionof the amplitudes A(β, µ) and B(β, µ), as presented inFig.1.The condition (6) is symmetric with respect to A andB, thus there is no need to specify which maximum am-plitude is sought. Moreover condition (6) does not de-pend on the specific choice of parameters, provided theyare independent. For instance, using the new parametersη and σ defined byβ = exp(η), µ = (σ − η) exp(2η), (7)the NST threshold condition becomes J(σ, η) = 0. Theinteresting point here is that for fixed σ, the new parame-ter η operates a shift of the solutions (5) u(x)→ u(x−η)and v(x) → v(x − η). Such a translation parameter al-ways exists in systems that possess a translation invari-ance on the whole x-axis.Generalizations. Generalization of the procedure toM -component systems is straightforward. To this end,let us denote the components by φm(x, z) = um(x)eiνmzand their amplitudes by um(0) = Am({ηn}), where ηn aretheM parameters of the solution (e.g. in the asymptoticsolution). Finding the NST threshold manifold amountsto setting to zero the determinant of the Jacobi matrixof the amplitudes with respect to the parameters, that isdet [Jmn] = 0, Jmn =∂Am∂ηn. (8)To illustrate our result with another interesting prob-lem, we may consider a multicomponent system having3soliton solutions, the (integrable) Manakov system [17],written here for spatial fields asi∂zψ + ∂2xψ + 2(|φ|2 + |ψ|2)ψ = 0,i∂zφ+ ∂2xφ+ 2(|φ|2 + |ψ|2)φ = 0. (9)It possesses the two-parameter soliton solution [18]ψ = eiz sin θ sech (x− η), φ = eiz cos θ sech (x− η). (10)Subject then, on the semi-infinite strip x > 0 and z ∈[0, L], to the boundary conditionψ(0, z) = Aeiz , φ(0, z) = Beiz , (11)and vanishing values as x → ∞, the Manakov systempossesses solution (10) provided the parameters (η, θ)are related to the driving amplitudes A and B byA = sin θ sech η, B = cos θ sech η . (12)The threshold curve in the (A,B) plane is then obtainedby solving J(η, θ) = 0 as given by (6). The solution is η =0, that is, in the variables (A,B), the circle A2+B2 = 1.This is illustrated on Fig.2 where we display a typicalsoliton formation and the NST threshold curve for whichthe points represent results of numerical simulations [13].0 0.5 100.51|φ||ψ|xz z5060700 10 20 0 10 20xgenerationBAevanescentwavegap solitonFIG. 2. (color online) Left: typical numerical simulation of(9) submitted to boundary conditions (11) with A = 0.45 andB = 0.9. The maximum amplitudes of the emitted solitonare |ψ|m ∼ 1.2 and |φ|m ∼ 2.3. Right: threshold curve in the(A,B)-plane where the dots result from numerical simulations[13] and the curve is the circle A2 +B2 = 1.Then one can check that the threshold manifold ofthe M -component Manakov system, i∂zφm + ∂2xφm +2(∑M1 |φn|2)φm = 0, subject to the boundary condi-tion φm(0, z) = Ameiz, is the M -dimensional sphere∑M1 A2m = 1.Parametric instability and supratransmission. As canbe seen from Figs.1 and 2, as soon as one of the ampli-tudes A or B crosses the threshold curve, the evanescentwave profile (soliton tail in integrable models) ceases toexist and gap solitons are emitted by the driven bound-ary. It turns out that, at least in the models we have in-vestigated, the NST threshold manifold also correspondsto values of the parameters around which the stabilityof the solutions changes. To substantiate this claim, weinvestigate why the conditions for parametric and NSTinstabilities might indeed be the same by performing lin-ear stability analysis of the stationary solutions (3) thatwe write in the formψ(x, z) = (u(x) + U(x, z))e2iz , (13)φ(x, z) =√α (v(x) + V (x, z))eiz . (14)where U and V are small perturbations that satisfy, ac-cording to (2), the boundary conditionU(0, z) = 0, V (0, z) = 0, (15)and vanishe at x→∞ for all z. Defining the real-valuedperturbation vectors Γr = (Ur, Vr)T and Γi = (Ui, Vi)T ,where index r (i) stands for real (imaginary) part, lin-earization of system (1) yields∂zΓr + P−Γi = 0, ∂zΓi − P+Γr = 0. (16)The matrix differential operators P± are given byP± =(α 00 1)(12∂2x − 2 2vv ∂2x − 1± u). (17)This is conveniently written as an eigenvalue problemby differentiating with respect to z. For the real partΓr(x, z) we may seek a solution Γr(x, z) = Φω(x) cos(ωz)and obtainP−P+Φω = ω2Φω (18)with boundary conditions (use ∂zΓi(0, z) = 0)Φω(0) = 0, (P+Φω)(0) = 0. (19)As P± and Φω are real valued, ω2 is also real valued.Then the solution is linearly stable when ω2 > 0 and un-stable for ω2 < 0, so that marginal instability is reachedat the bifurcation point ω = 0 providing the parametricinstability threshold.An essential property of the operator P+, obtained bydifferentiation of (4) with respect to the parameters isP+ ∂∂η(uv)= 0, P+ ∂∂σ(uv)= 0. (20)Thus a two-parameter family of solutions of (18) at ω = 0readsΦ0(x) = a∂∂η(uv)+ b∂∂σ(uv), (21)for arbitrary constants (a, b) ∈ R2. Though it is not themost general solution of (18), it seems to be the only oneable to satisfy the boundary conditions (19). Requiringthen Φ0(0) = 0, with u(0) = A and v(0) = B, eventu-ally yields J(η, σ) = 0, namely the parametric instability4condition (6). Thus the NST threshold condition (6),i.e., the condition for the maximum allowed amplitudesA and B at the boundary x = 0, actually coincides withthe onset of instability of the solution for the correspond-ing critical values of the parameters (here η and σ).To check this statement, we have computed numeri-cally the eigenvalue ω2 of the differential equation (18)around the bifurcation point ω = 0 by varying η and σaround their critical values ηc and σc defined by the solu-tion of J(η, σ) = 0. The result is plotted in Fig.3 wherewe have used the asymptotic series solution (u, v) at or-der N = 16 as previously. As can be seen from the figuremarginal instability is actually reached at the criticality(ηc and σc) when ω crosses zero.–0.100.1–0.1 0.1–0.100.1–0.1 0.1ω2η−ηc σ−σccσ=σ cη=ηω2−0.1−0.1−0.0.10.1−0.1 0 0 0.10.1FIG. 3. Plots of the eigenvalue ω2 in terms of the parameters(η, σ) as a function of η− ηc for σ = σc (left) and function ofσ − σc for η = ηc (right) again with α = 1.1.Comments and conclusion. The method presentedhere can be readily applied to the simple case of the scalarnonlinear Schro¨dinger equation i∂zψ+ ∂2xψ+2|ψ|2ψ = 0to get interesting insight about the occurrence of NST.It is found that (i) the asymptotic series solution actu-ally sums up exactly to furnish the one-soliton solution,(ii) the fundamental parameter is the position η of thesoliton maximum, (iii) the threshold is indeed the max-imum amplitude of this static soliton reached at η = 0,(iv) the variations of the eigenvalue ω2 around zero isgiven by ω2 = −4η + o(η) that, straightforwardly, givesthe marginal instability threshold η = 0.In such single component systems as NLS, the instabil-ity occurs always at the maximum amplitude of the solu-tion [19]. On the contrary, the solution of the D3Wmodeldoes not display any maximum nor any other geometricevidence that the NST threshold has been reached.In conclusion, we have solved the challenging problemof determining the threshold for nonlinear supratrans-mission in nonintegrable N -component systems. This isobtained in two steps: first by deriving an asymptoticsolution based on their linear evanescent profile that de-pends on N parameters, and second by solving Eq. (8).In the parameter space the latter condition results in aN − 1 dimensional manifold that determines the changeof stability of the (asymptotic) solution. Expressed interms of the amplitudes, it gives rise to the sought NSTthreshold. The situation is highly simplified in the caseof an integrable system, or a system that possesses an ex-act static solitonlike solution, since one can work directlywith the solution to obtain the threshold.Finally, since no analytical expression is required, themethod can be successfully applied to a wide class ofnonintegrable nonlinear multicomponent systems.Work done as part of the programme GDR 3073PhoNoMi2 (Photonique Nonline´aire et Milieux Micro-structure´s).∗ Permanent address: Laboratoire de Physique des Lasers,Atomes et Mole´cules, CNRS-INP-UMR8523,Universite´des Sciences et Technologies de Lille, 59655 Villeneuved’Ascq, (France)[1] F. Geniet, J. Leon, Phys Rev Lett 89 (2002) 134102[2] P. Maniadis, G. Kopidakis, S. Aubry, Physica D 216(2006) 121[3] J. E. Macias-Daz, A. Puri, Phys Lett A 366 (2007) 447[4] J. Leon, A. Spire, Phys Lett A 327 (2004) 474[5] D. Taverner, N.G.R. Broderick, D.J. Richardson, R.I.Laming, M. Ibsen, Opt Lett 23 (1998) 328[6] R. Khomeriki, Phys Rev Lett 92 (2004) 063905[7] J. Leon, Phys Rev E 70 (2004) 056604[8] J. Leon, Phys Lett A 319 (2003) 130[9] H. Susanto, SIAM J Appl Math 69 (2008) 111[10] R. Khomeriki, J. Leon, Phys Rev Lett 94 (2005) 243902[11] See e.g. the review: A.V. Buryak, P. Di Trapani, D.V.Skryabin, S. Trillo, Phys. Rep. 370 (2002) 63-235[12] V.A. Brazhnyi, V.V. Konotop, S. Coulibaly, M. Taki,CHAOS 17 (2007) 037111[13] The points are obtained as follows: at fixed B no solitonis emitted for some A while atA+ǫ solitons are generated.The point is then set as (A+ ǫ/2, B) and ǫ can be madeas small as needed within the accuracy of the numericalcode. As component profiles are initially set to zero, wesmoothly increase amplitudes A and B from zero up totheir respective assigned value to avoid shocks. This iswhy simulations are only displayed from z = 60 on.[14] The driving wavenumber is scaled to any value by usingthe invariance D3W under the transformation z → kz,x→√k x and (ψ, φ, δ)→ (ψ,φ, δ)/k.[15] G. Leo, G. Assanto, Opt. Lett. 22 (1997) 1391[16] M.T.G. Canva, R.A. Fuerst, S. Baboiu, G.I. Stegeman,G. Assanto, Opt. Lett. 22 (1997) 1683[17] S.V. Manakov, Sov Phys JETP 38 (1974) 248[18] The driving wavenumber is scaled to k = 1 by usingthe invariance of the system under the transformationz → kz, x→√k x and (ψ, φ)→ (ψ, φ)/√k.[19] The NST condition (8) for a one-component system leadsto ∂A/∂η = 0 that is to (∂u/∂x)(x = 0) = 0.

Nonlinear supratransmission in multicomponent systems

Nonlinear supratransmission in multicomponent systems

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