Perturbation theory for the modified nonlinear Schr¨odinger solitons V.S. Shchesnovicha and E.V. Doktorovb aDivision for Optical Problems in Information Technologies, National Academy of Sciences of Belarus, Zhodinskaya St. 1/2, 220141 Minsk, Republic of Belarus bB.I.Stepanov Institute of Physics, 68 F. Skaryna Ave., 220072 Minsk, Republic of Belarus 8 9 9 Abstract 1 The perturbation theory based on the Riemann-Hilbert problem is developed for the n modified nonlinear Schr¨odinger equation which describes the propagation of femtosecond a J optical pulses in nonlinear single-mode optical fibers. A detailed analysis of the adiabatic approximation to perturbation-induced evolution of the soliton parameters is given. The 4 1 linear perturbation and the Raman gain are considered as examples. 1 v PACS. 03.40Kf - Waves and wave propagation: general mathematical aspects. 02.30Jr - 6 Partial differential equations. 1 0 Keywords: soliton perturbation theory, femtosecond soliton, modified nonlinear Schr¨odinger 1 equation, the Riemann-Hilbert problem. 0 8 1. Introduction 9 / t The study of the dynamical processes associated with the propagation of high-power optical pulses in n single-mode nonlinear fibers is based as a rule on the integrable nonlinear Schr¨odinger equation(NLSE) [1]. i - Various realistic effects accompanying the soliton propagationand destroying the integrability of the NLSE v are usually treated as perturbations. There are different approaches to describe analytically perturbation- l o induced dynamics of the NLS solitons [2-8], for review see Ref. 9. It is evident that the ”quality” of s taking into account for the above effects depends crucially on smallness of a parameter responsible for a : v definite perturbation. Just this situation takes place with the Kerr nonlinearity dispersion effect. Being i X sufficiently small in the picosecond pulse duration region, it becomes essential for femtosecond solitons, having a parameter of the order 10−2 10−1. Hence, streactly speaking, this effect cannot be treated as a r a perturbation for the femtosecond regio−n of soliton pulse duration. 1 The natural approach to treat analytically the dynamics of femtosecond solitons is to consider the so- called perturbed modified nonlinear Schr¨odinger equation (MNLSE) [10] 1 iq + q +iα q 2q + q 2q =r, (1) z 2 ττ | | τ | | (cid:0) (cid:1) where the term with the real parameter α governs the effect of the Kerr nonlinearity dispersion (self- steepening) and r accounts for small effects which we will consider as perturbation. Here q(τ,z) is the normalizedslowlyvaryingamplitude ofthe complexfieldenvelope,z isthe normalizedpropagationdistance along the fiber, τ is the normalized time measured in a frame of reference moving with the pulse at the group velocity (the retarded time). It is remarkable that MNLSE (1) with zero r.h.s. is still integrable by the inverse scattering transform (IST) method [11], though the linear spectral problem associated with the MNLSE is different from that for the NLSE. Our primary goal is to develop a simple formalism to treat analytically the femtosecond soliton dynam- ics governed by Eq. (1). Three points should be stressed which differ our approach from the previously known ones. First, we accountfor the Kerr nonlinearity dispersion effect exactly. In other words, we do not make any hypothesis about smallness of α in Eq. (1). Moreover, we consider as a background solution not the sech-like pulse of the NLS type but precisely the MNLS soliton. Finally, the third point is relevant to the formalism, namely, we employ the Riemann-Hilbert (RH) problem, which was provedto be effective for treatingperturbationstononlinearevolutionequationsintegrablebymeansoftheZakharov-Shabatspectral problem[12-15]. Recently,we developedthe RH problem-basedapproach[16-18]for solvingnonlinearequa- tions integrable by the Wadati-Konno-Ichikawa spectral problem [11]. This approach includes the MNLSE andsomeits generalizations. Thisdevelopmentservesasabasefor takingintoaccountsmallperturbations. The paper is arranged as follows. In Sec. 2 the basic results on the RH problem-based approach to the MNLSE is summarized. In Sec. 3 the general one-soliton solution of the MNLSE is derived in a form which, we believe, is as simple as possible. Here we also discuss the limiting transition to the NLS soliton. In Sec. 4 we obtain the perturbation-induced evolution equations for the RH problem data related to the solitonparametersanddiscusspeculiaritiesoftheperturbationtheoryforgaugeequivalentequations. Sec.5 is devoted to the adiabatic approximation. Here we consider as an example the linear perturbation (excess gain or fiber loss) and the Raman self-frequency shift [19-21]. Concluding remarks are contained in the last section. 2. Riemann-Hilbert problem for MNLSE In this section we summarize the basic results concerning the approachto the MNLSE based on the RH probem. Let us write the MNLSE in the general form 1 iq + q +iα q 2q +β q 2q =0, (2) z 2 ττ | | τ | | (cid:0) (cid:1) 2 where α and β are real parameters. Eq. (2) is integrable via the IST method and can be represented as the compatibility condition U V +[U,V]=0 for the following system of two linear matrix equations z τ − Φ =Λ(k)[σ ,Φ]+2ikQΦ UΦ Λ(k)Φσ , (3a) τ 3 3 ≡ − 4i iβ Φ =Ω(k)[σ ,Φ]+ k3Q+2ik2Q2σ kQ+kQ σ 2iαkQ3 Φ z 3 3 τ 3 α − α − (cid:18) (cid:19) VΦ Ω(k)Φσ . (3b) 3 ≡ − 0 q Here the Hermitian matrix Q = represents the potential of the linear spectral problem (3a), q 0 (cid:18) (cid:19) Λ(k)= (2i/α) k2 β/4 , Ω(k)= (4i/α2)(k2 β/4)2, the bar stands for complex conjugationand k is a − − − − spectral parameter. (cid:0) (cid:1) As seen from Eq. (3a), the MNLSE (2) belongs to the class of equations integrable by means of the Wadati-Konno-Ichikawa spectral problem [11]. However, Eq. (2) is not canonical among the equations of this class [16]. The canonical equation 1 iq′ + q′ iαq′2q′ +β q′ 2q′+α2 q′ 4q′ =0 (4) z 2 ττ − τ | | | | does not admit as obvious a physical interpretation as the MNLSE, but possesses the Lax representation too, Φ′ =Λ(k)[σ ,Φ′]+ 2ikQ′+iαQ′2σ Φ′ U′Φ′ Λ(k)Φ′σ , (5a) τ 3 3 ≡ − 3 (cid:16) (cid:17) 4i iβ α Φ′ =Ω(k)[σ ,Φ′]+ k3Q′+2ik2Q′2σ kQ′+kQ′σ + [Q′,Q′] z 3 α 3− α τ 3 2 τ iα2 + Q′4σ Φ′ V′Φ′ Ω(k)Φ′σ , (5b) 3 3 2 ! ≡ − 0 q′ where Q′ = . The spectral problem (5a) associated with Eq. (4), as distinct from the spectral q′ 0 (cid:18) (cid:19) problem (3a), is compatible with the canonical normalization condition Φ′(k = ) = I, where I is the ∞ 2 2 identity matrix. Eqs. (2) and (4) are gauge equivalent equations interrelated by the following gauge × transformation: Q=g−1Q′g, (6) whereg(τ,z)=Φ′(k =0). Thereby,solutionsoftheMNLSEcanbeobtainedfromthoseofEq.(4)bymeans of simple algebraic transformation(6). The RH problem formalismcan be developed equivalently for either 3 the MNLSE (2) or Eq. (4), the RH problem data being invariant under the gauge transformation. Because the formulation of the RH problem with the canonical normalization condition has a number of technical advantages in calculation of soliton solutions, we will develop the RH formalism for Eq. (4) and give the transition relations to the MNLSE. To construct the RH problem associated with Eq. (4), consider the matrix Jost-type solutions J′ of ± Eq. (5a) which satisfy the asymptotic conditions J′ I at τ . By the standard analysis of the ± → → ±∞ Volterra-type integralequations for J′ which follow from Eq. (5a) and the above asymptotic properties,we ± conclude that the following matrix function ((J′ ) means the l-th column of J′ ) ± ·l ± Φ′ (k)= (J′ ) (k), (J′ ) (k) , (7) + + ·1 − ·2 (cid:18) (cid:19) being a solution of Eq. (5a), is holomorphic in the two quadrants of the complex k-plane which are defined by the condition αRe(k)Im(k) 0. The scattering matrix S′(k) is defined in the usual way: ≤ J′ E =J′ ES′, E exp(Λ(k)σ τ). (8) − + ≡ 3 Note that detS′ =1 due to detJ′ =1. The Zakharov-Shabatfactorization [22] of the scattering matrix, ± 1 S′ (S′−1) 0 S′ =S′S′ , S′ = 12 , S′ = 11 , (9) + − + 0 S′ − (S′−1) 1 (cid:18) 22 (cid:19) (cid:18) 21 (cid:19) allows us to represent Φ′ (k) in two equivalent forms: + Φ′ =J′ ES′ E−1 =J′ ES′ E−1. (10) + + + − − Since the potential Q′ is Hermitian, we have the following identities: J′† (k)=J′−1(k), S′†(k)=S′−1(k), k k :Re(k)Im(k)=0 . ± ± ∈{ } Hermiticityofthe potentialalsoenablestodefinethe matrixfunctionconjugatedtoΦ′ (k)andholomorphic + in the rest two quadrants of the complex k-plane, i.e., which are given by the condition αRe(k)Im(k) 0: ≥ t Φ′ −1(k)=Φ′ †(k)= (J′ −1) (k), (J′ −1) (k) , (11) − + + 1· − 2· (cid:18) (cid:19) where (J′ −1) denotes the l-th row of the matrix J′ −1 and superscript t means transposition. The linear ± l· ± spectral problem (5a) possesses the parity symmetry [16]. It can be summarized to the following important identities: J′ (k)=J′ (k), S′(k)=S′(k), S′ (k)=S′ (k), P ± ± P P ± ± 4 where is the parity operator defined by F(k)= F(d)( k) F(off)( k) σ F( k)σ , F(d) and F(off) 3 3 P P − − − ≡ − are diagonal and off-diagonal parts of a matrix F. These identities give Φ′ (k), k k:αRe(k)Im(k) 0 , Φ′(k)=Φ′(k), Φ′(k)= + ∈{ ≤ } (12) P Φ′ (k), k k:αRe(k)Im(k) 0 . (cid:18) − ∈{ ≥ } (cid:19) Here Φ′(k) is a matrix function piecewise meromorphic in the complex k-plane and discontinuous through the curve k k:Re(k)Im(k)=0 . ∈{ } Letus returnto the gaugeequivalencebetween Eqs.(2)and (4). FromEq.(12)takenatk =0it follows thatthe gaugetransformationmatrixg isdiagonal,whileEqs.(5a)and(10)leadtothe followingexpression for g: ∞ exp iα dτ q′ 2 0 − | | g ≡Φ′(k =0)= (cid:18) 0Rτ (cid:19) exp iα τ dτ q′ 2 . (13)  − | |   (cid:18) −∞ (cid:19)   R  The matrix function Φ(k) which results from Φ′(k) by means of the transformation Φ(k)=g−1Φ′(k), (14) is a solution of the linear problem (3a), possesses the same conjugation and meromorphic properties as Φ′(k) does, and is an eigenfunction of the parity operator too. For the linear problem (3a), the Jost-type P solutions J (k), the scattering matrix S(k), and its factorization S (k) are constructed similarly to the ± ± problem (5a). The matrix function Φ(k) is given through the Jost-type solutions and the factorization by the the same formulas as Φ′(k) does (see Eq.(10)). The following relations are valid J =g−1J′ g , S =g−1S′g , S =g−1S′ , (15) ± ± ± + − ± ± ± where g = lim g. ± τ→±∞ Now we can formulate the RH problem associated with Eq. (4). Indeed, using Eqs. (7) and (11) as well as the relation E(k)=E(k) for k k :Re(k)Im(k)=0 , we write (S′ (k) S′ (k)) ∈{ } 12 ≡ 12 Φ′ −1(k)Φ′ (k)=E(k)S′ †(k)S′ (k)E−1(k) − + + + 1 S′ (k) =E(k) ′ 12 E−1(k) G(k), (16a) S (k) 1 ≡ (cid:18) 12 (cid:19) Φ′(k) I, k , (16b) → →∞ 5 where k k:Re(k)Im(k)=0 . It is a problem of analytic factorization of the nondegenerate matrix G(k) ∈{ } given on the contour defined by the following disconnected oriented set: C =sgn(α) ( ,0),(0,0) (0,0),( i ,0) α { ∞ }∪{ − ∞ } (cid:18) ( ,0),(0,0) (0,0),(0,i ) ∪{ −∞ }∪{ ∞ } (cid:19) of the k-plane axes. Here sgn(α) means bypassing the contour in the reverse direction for α < 0. The functions Φ′ (k) are just a solution of the RH problem (16). The uniqueness of this solution is provided by ± the canonical normalization condition (16b). Substituting the asymptotic decomposition of Φ′(k) in the inverse power series of k, Φ′(k)=I+k−1Φ′[1]+..., into the spectral equation (5a) and taking advantage of Eq. (12), we reconstruct the potential Q′: 1 1 Q′ = [σ ,Φ′[1]]= lim k[σ ,Φ′(k)]. (17) 3 3 α αk→∞ Hence, to solve the MNLSE (2), we should at first solve the RH problem (16), then obtain the potential Q′ (17) and transform it by Eq. (6). It should be noted that the RH problem (16) remains unchanged under the gauge transformation (14), except for the normalization of the matrix Φ(k) at infinity, namely, Φ(k) g−1 at k . In→general, th→e fu∞nctions detΦ′ (k) and detΦ′ −1(k) have zeros in their regions of analiticity, the RH + − problem being said to be nonregular (or with zeros). It follows from Eq. (11) that zeros of the above determinants are complex conjugate, while the parity symmetry (Eq. (12)) tells us that zeros appear by pairs, i.e., detΦ′ ( k ) = 0 for the j-th zero k . The solution of the RH problem with zeros can be + ± j j factorized [23] Φ′ (k)=Φ′ (k)Γ(k), detΦ′ (k)=0 (18) ± o± o± 6 by means of the solution Φ′ of the regular RH problem: o± (Φ′ )−1(k)Φ′ (k)=Γ(k)G(k)Γ−1(k), (19a) o− o+ Φ′(k) I, k , (19b) o → →∞ which is posed on the same contour C . The matrix function Γ(k) represents the contribution of zeros. In α the case of the single pair of zeros k k this function is given by (see also Ref. 18) 1 ±1 ± ≡ p D−1 p p D−1 p Γ=I | ji jl h l |, Γ−1 =I+ | ji jl h l |, (20) −j,Xl=±1 k−kl j,Xl=±1 k−kj 6 where D =(k k )−1 p p , D−1 (D−1) , p p =(p ) (p ) +(p ) (p ) , vector-rows p are ln n− l h l | ni ·· ≡ ·· h l | ni l 1 n 1 l 2 n 2 h ±1 | related to vector-columns p by conjugation, i.e., p = p †, and the latter are given by ±1 ±1 ±1 | i h | | i Φ′ (k ) p =0. (21) + ±1 | ±1i The parity symmetry immediately gives a relation between p and p . Indeed, the identity Φ′ (k ) | −1i | +1i + 1 | p = σ Φ′(k )σ p = 0 leads, due to uniqueness of p , to the relation p = σ p . From +1 3 −1 3 +1 −1 −1 3 +1 Eq.i(11) it follows th|at pi Φ′ −1(k )=0. This identity| andiEq. (21) ensure t|hat tihe mat|rix fuinctions h ±1 | − ±1 Φ′ (k) and (Φ′ )−1(k) are holomorphic in the respective quadrants of the complex k-plane. o+ o− In the general case of the RH problem with N pairs of zeros of detΦ′ (k) the matrix function Γ(k) is + given by Eq. (20) with j,l N, N +1,..., 1,1,...N 1,N and k k . −s s ∈{− − − − } ≡− 3. Soliton solution of MNLSE The RH problem data are divided into two parts: discrete data k , j { p ,j = 1,...,N (2N is the whole number of zeros of detΦ′ (k)) and continuous datum G(k). Soliton | +ji } + solutionscorrespondtotheRHproblemwithzerosprovidedG(k)=I,i.e.,Φ′ (k)=I. Inotherwords,only o± the matrix Γ(k) is responsible for solitons. We will consider the simplest case of one-soliton solution of the MNLSE (2). We have from Eqs. (6) and (17): 1 Q= Γ−1(k =0) lim k[σ ,Γ(k)]Γ(k =0), (22) 3 α k→∞ wherethematricesΓ(k)andΓ−1(k)aregivenbyEq.(18)anditistakenintoaccountthatg =Γ(k =0). As regardsthecoordinatedependenceofthevector-column p ,itisdeterminedbydifferentiationofEq.(21) +1 | i withrespecttoτ andz andtakingadvantageofEqs.(5). ItshouldbenotedthatEq.(21)determines p +1 | i only up to an arbitrary norm. We obtain in a particular case p =Λ(k )σ p , p =Ω(k )σ p . (23) +1 τ 1 3 +1 +1 z 1 3 +1 | i | i | i | i Integration of the above equations gives po p =efσ3 po , po = 1 , | +1i | i | i po (cid:18) 2 (cid:19) where f =Λ(k )τ +Ω(k )z and po is an integrationconstant determined up to an arbitrarynorm. Let us 1 1 | i define po/po =exp(a+iϕ), where a and ϕ are real constants. Ultimately, we have 1 2 2 exp(a+iϕ+f) 2i β 4i β p = , f = k2 τ k2 z. (24) | +1i e−f −α 1− 4 − α2 1 − 4 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) 7 Substituting Eq. (24) into Eq. (20), we obtain for Γ(k) the following expression 2ea k ex D−1 +D−1 keiψ D−1 +D−1 Γ(k)=I 1 ++ −+ ++ −+ . (25) − k2−k21 (cid:18) ke−iψ(cid:0)D+−+1 −D−−+1(cid:1) k1e−x(cid:0) D+−+1 −D−−+1(cid:1) (cid:19) (cid:0) (cid:1) (cid:0) (cid:1) Here 8ξη 4 β x=a+2Ref =a τ + ξ2 η2 z , − α α − − 4 (cid:18) (cid:20) (cid:21) (cid:19) (26) 4 β 2 β 2 ψ =ϕ+2Imf =ϕ ξ2 η2 τ + ξ2 η2 4ξ2η2 z , − α((cid:18) − − 4(cid:19) α (cid:20) − − 4(cid:21) − ! ) and k = ξ iη (due to the condition αRe(k )Im(k ) < 0 we will have ξ > 0 and η > 0 for α > 0). From 1 1 1 − the definition D =(k k )−1 p p we derive the following properties of the matrixD: D = D , ln n l l n ++ −− − h | i − D = D . Hence, −+ +− − D−1 D−1 = (D D )(detD)−1 =(D D )−1 ++± −+ − ++± −+ ++∓ −+ i 1 −1 =e−a chx shx . (27) η ∓ ξ (cid:18) (cid:19) Now,wehavealltocalculateone-solitonsolutionq oftheMNLSE(2). Substituting Eqs.(26)and(27)into s Eq. (25), then, in its turn, Eq. (25) into Eq. (22) we obtain ξη k e−x+k ex q Q =8i 1 1 eiψ. (28) s 12 ≡ α (k ex+k e−x)2 1 1 This is a generalformofthe one-solitonsolutions to the MNLSE (2) whichdepends onfour realparameters ξ, η, a, and ϕ. The solution (28) is written in terms of the coordinates x and ψ (26) comprising linear combinations of the normalized retarded time τ and distance along the fiber z. It should be stressed that α enters the denominator of the soliton solution (28). In other words, we account nonperturbatively for the pusleself-steepeningeffect. SimilarlytotheNLSE,theparametersaandψplaytheroleoftheinitialposition and phase, respectively, while the other two parameters ξ and η do not admit so obvious interpretation. In anycase,weseefromEq.(28)thatthenormalizedhalf-widthw andvelocityv ofthe solitonarerepresented by α β 4 w = , v = (ξ2 η2). (29a) 2ξη α − α − 8 As regards the soliton amplitude A, it is natural to admit it in the following form: 4ξη A= . (29b) α(ξ2+η2)21 The MNLS soliton (28) has a number of peculiarities which distinct it form the NLS soliton. First, the MNLS soliton has nonzero phase difference at its limits. Indeed, k e−x+k ex k /k2, x , 1 1 1 1 →∞ 2 (k1ex+k1e−x)2 −→( k1/k1, x→−∞ and the said phase difference reads ∆ψ =arg(q (z )) arg(q (z ))=6arg(k ). s s 1 →∞ − →−∞ Further, the integral of the soliton amplitude ∞dτ q = π ∞ (41)n 2 2ξη 2n, | s| 2(ξ2+η2)12 n! ξ2+η2 −Z∞ nX=0(cid:18) (cid:19) (cid:18) (cid:19) where 1 1 +1 1 +2 ... 1 +n 1 , depends on the parametersξ and η, in contrastto the same 4 n ≡ 4 4 · · 4 − integral of the NLS soliton which does not depend on the soliton parameters. Moreover, the important (cid:0) (cid:1) (cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1) invariant of Eq. (2), namely, the number of particles or the optical energy of the soliton, ∞ 4 π E = dτ q 2 = arg(k ), 0< arg(k ) < , s 1 1 | | α | | 2 Z −∞ has the upper limit 2π/α. The phase difference and optical energy of the MNLS soliton are related: | | ∆ψ = 3Eα/2. − The above properties of the MNLS soliton (28) resemble those of the dark NLS soliton, which also has nonzero phase difference and a relation between its energy and the phase difference [24]. Nevertheless, the soliton(28)withβ >0reducestoabrightNLSsolitonatα 0. Tocarryoutthislimitoneshouldtakeinto → accountthatthe Laxpairforthe NLSEshouldbe producedatα 0fromthe Laxpair(3)forthe MNLSE. → This condition implies that the spectral parameter k depends on α and gives the following prescription: 2(k2 β/4)/α k at α 0 or MNLS − →− NLS → 2 ξ2 β − 4 4ξη ξ , η , α 0, (cid:16) α (cid:17) →− o α → o → 9 where k =ξ iη and k =ξ +iη In other words, we have the decomposition MNLS NLS o o − √β α α ξ = ξ +O(α2), η = η +O(α2), α 0, (30) o o 2 − 2√β 2√β → which transforms the MNLS soliton (28) to the NLS soliton q =(2iη /√β)eiψosechx , where ψ =ϕ+2ξ τ 2(ξ2 η2)z, x =a 2η (τ 2ξ z). s o o o o − o − o o − o − o 4. Perturbation-induced evolution of RH problem data Followingourstrategy,tofindcorrectiontothe solitonsolutionoftheMNLSEcausedbyaperturbation, we should at first derive the perturbation-induced evolution equations for the RH problem data. Decompo- sition of these equations in the asymptotic power series with respect to the perturbation will produce the consequent corrections to the soliton solution. In Sec. II we have seen that the gauge equivalent Eqs. (2) and (4) have resembling IST schemes with simple mutual relations (15). As the RH problem data are invariant under the gauge transformation (6), we can choose between the two IST formulations the most convenient one for calculation of corrections to soliton solutions. Consider eq. (2) with a small perturbation at the r.h.s. of the equation, i.e., Eq. (1). The perturbation causes a variation δU of the potential, what, in its turn, leads to a variation δJ of the ± Jost-type solutions. From the spectral problem (3a) we obtain the following equation for δJ ± (δJ ) = Λ(k)δJ σ +δUJ +UδJ , lim J−1δJ =0. ± τ − ± 3 ± ± τ→±∞ ± ± (cid:0) (cid:1) Therefore, τ δJ =J E dτE−1J−1δUJ E E−1, (31) ± ±  ± ±  Z ±∞ where δU = (δU/δz)δz. It should be stressed that the same representation of the variation δJ′ of the ± Jost-type solutions to the spectral problem (5a) follows from Eq. (31) by the simple substitutions U U′ → and J J′ and such a procedure can be carried out at any step below. ± → ± Now we introduce a useful matrix function τ δU γ( ,τ)= dτE−1Φ−1 Φ E. (32) ±∞ + δz + Z ±∞ Then the variation derivative δJ /δz takes the form ± δJ ± =J ES γ( ,τ)S−1E−1. (33) δz ± ± ±∞ ± 10