Exactly solvable model for nonlinear pulse propagation in optical fibers

EXA CTL Y SOL V ABLE MODEL F OR NONLINEAR PULSE PR OP A GA TION IN OPTICAL FIBERS JONA T AN LENELLS Abstract. The nonlinear Sc hr¨ odinger (NLS) equation is a fundamen tal mo del for the nonlinear propagation of ligh t pulses in optical fib ers. W e consider an in tegrable generalization of the NLS equation which was first deriv ed by means of bi-Hamiltonian methods in [A. S. F ok as, Phys. D 87 (1995), 145–150]. The purp ose of the presen t pap er is threefold: (a) W e sho w ho w this generalized NLS equation arises as a mo del for nonlinear pulse propagation in monomode optical fib ers when certain higher-order nonlinear effects are taken in to account; (b) W e sho w that the equation is equiv alent, up to a simple c hange of v ariables, to the first negative member of the integrable hierarch y asso ciated with the deriv ative nonlinear Sc hr¨ odinger equation; (c) W e analyze tra veling-w a v e solutions. P ACS numbers (2008) : 42.81.Dp, 02.30.Ik. Keywords : Generalized nonlinear Sc hr¨ odinger equation, optical fiber, soliton. 1. Introduction The nonlinear Schr¨ odinger (NLS) equation (1.1) iu t + γ u xx + ρ | u | 2 u = 0 , x ∈ R , t > 0 , where γ and ρ are real parameters and u ( x, t ) is a complex-v alued function, is a fundamen tal model for the nonlinear propagation of ligh t pulses in optical fib ers [1]. Equation (1.1) is a completely integrable system and can b e linearized b y the in v erse scattering transform formalism [2]. In the context of fib er optics, the soliton solutions of (1.1) are of particular in terest: b ecause of their inheren t stabilit y , solitons can trav el unchanged ov er long distances, making them ideal carriers of information in optical transmission systems. In this paper w e consider a generalization of the NLS equation, whic h also admits soliton solutions, and whic h, as we shall argue, mo dels the propagation of nonlinear ligh t pulses in optical fib ers when certain higher-order nonlinear effects are tak en into accoun t. The equation (1.2) iu t − ν u tx + γ u xx + ρ | u | 2 ( u + iν u x ) = 0 , x ∈ R , t > 0 , where ν, γ , ρ are real parameters and u ( x, t ) is a complex-v alued function, w as deriv ed in [3] as an equation with tw o distinct, but compatible, Hamiltonian form ulations. Just lik e the bi-Hamiltonian structure of the w ell-kno wn Kortew eg-de V ries equation can b e p erturb ed to yield the in tegrable so-called Camassa-Holm equation [4], the same mathematical tric k applied to the tw o Hamiltonian op erators asso ciated with the NLS equation yields equation (1.2)—note that (1.2) reduces to (1.1) when ν = 0. Equation (1.2) admits a Lax pair form ulation and the initial v alue problem can b e solv ed by means of the in v erse scattering transform [5]. Date : Octob er 29, 2018. The author thanks Matz Lenells for v aluable discussions and ac knowledges supp ort from a Marie Curie In tra-Europ ean F ello wship. 1 2 JONA T AN LENELLS F or large classes of dispersive nonlinear PDE’s, the NLS equation arises asymptot- ically as the equation satisfied at first appro ximation b y the amplitude A of a slo wly mo dulated mono c hromatic w av e [6]. It w as demonstrated in [3] that an application of asymptotic techniques to a KdV type equation gives rise to an equation of the form (1.2) provided that one retains terms of the next asymptotic order b ey ond those included in the case of the NLS equation. In this sense equation (1.2) is a general- ization of the NLS equation. Ho wev er, to the b est of our knowledge a more direct ph ysical deriv ation of (1.2) has not yet been presen ted. The purp ose of the present pap er is threefold: (a) W e show how equation (1.2) arises as a mo del for nonlinear wa v e propagation in monomo de optical fib ers; (b) W e sho w that (1.2) is equiv alent, up to a simple change of v ariables, to the first negativ e mem b er of the integrable hierarch y associated with the deriv ative nonlinear Sc hr¨ odinger (DNLS) equation; (c) W e analyze tra v eling-w a v e solutions. Let us comment on (a), (b), and (c). (a) Physical deriv ation. The propagation of nonlinear pulses in optical fib ers is describ ed to first order by the NLS equation (1.1). Ho wev er, when considering very short input pulses it is necessary to include additional terms accoun ting for higher- order effects suc h as self-steep ening, Raman scattering, and third-order disp ersion [7, 8]. While the rmore general esulting equation is not in tegrable, there exist exactly solv able appro ximations which capture v arious asp ects of the higher-order phenom- ena. Most notably , the DNLS, Hirota, and Sasa-Satsuma equations are all in tegrable reductions of the more general equation and are used as mo dels when studying wa ve propagation for ultrashort input pulses [9, 7, 10]. Although eac h of these reduc- tions ignores some of the important nonlinear effects, their solutions can b e used as a first appro ximation when p erforming a more complete analysis in v olving perturbation metho ds. Moreov er, the p ossibilit y of presenting exact solutions to these models mak e them v aluable to ols for understanding the ph ysical influence of the v arious terms. After recalling certain asp ects of the standard deriv ation of the NLS equation in nonlinear fib er optics, we indicate in Section 2 ho w equation (1.2) appears when taking into accoun t terms that are normally ignored. (b) Relation to DNLS. The deriv ative nonlinear Sc hr¨ odinger equation (1.3) iq t + q xx + σ i ( | q | 2 q ) x = 0 , σ = ± 1 , and gauge transformed v ersions thereof, ha v e sev eral applications in plasma physics and nonlinear fib er optics [11, 10]. It was sho wn in [12] that (1.3) admits a Lax pair formulation and can b e solved b y means of in verse scattering techniques. Be- ing integrable, equation (1.3) admits an infinite num b er of conserv ation la ws and a bi-Hamiltonian formulation. Moreo ver, asso ciated with equation (1.3) is an infinite hierarc h y of equations generated by the bi-Hamiltonian structure, equation (1.3) be- ing the second p ositiv e member. In Subsection 3.2 we demonstrate that, up to a simple change of v ariables, equation (1.2) is nothing but the first negativ e member of this hierarch y giv en by (1.4) u tx = u − iσ | u | 2 u x , σ = ± 1 . Since the parameter σ in (1.4) can b e remov ed by the replacement ( t, x ) → ( σ t, σ x ), this sho ws that the parameters ν, γ , ρ in equation (1.2) can all b e set to 1 b y a c hange of v ariables. In particular, in con trast to the case of the NLS equation whic h comes in a fo cusing as w ell as in a defo cusing v ersion dep ending on the v alues of the parameters, and solitons only exist in the the focusing regime, all v ersions of equation NONLINEAR PULSE PR OP AGA TION IN OPTICAL FIBERS 3 (1.2) are mathematically equiv alent up to a c hange of v ariables. This observ ation is consisten t with the fact that solitons for (1.2) exist for all v alues of ν, γ , ρ [5]. Our inv estigation of the relationship b etw een (1.2) and (1.3) was motiv ated by the observ ation that if u x is identified with q , then the Lax pairs asso ciated with equations (1.2) and (1.3) ha v e iden tical x -parts [5]. Let us point out that a similar link exists b et w een the KdV and Camassa-Holm equations: The Camassa-Holm equation is deriv ed mathematically from the KdV equation in an analogous manner that (1.2) is derived from the NLS equation. Moreo v er, it is known that the Camassa-Holm equation is related by a (nonlo cal) change of v ariables to the first negative mem ber of the KdV hierarc h y [13]. Since the gauge transformation relating (1.2) to (1.3) is a lo cal change of v ariables, the link in the presen t case is more direct. (c) T rav eling wa ves. In Section 4 we consider tra veling-w av e solutions of (1.2). F or simplicit y , w e consider the simpler, but equiv alen t, equation (1.4). If w e assume that the solution u ( x, t ) of (1.4) has the sp ecial form (1.5) u ( x, t ) = ϕ ( x − ct ) e i ( kx − Ω t + θ ( x − ct )) , where k , Ω, c are real parameters and ϕ and θ are real-v alued functions, the partial differen tial equation for u reduces to a first-order ODE for ϕ . A straigh tforw ard anal- ysis of this equation yields the qualitative structure of large classes of tra v eling-wa ve solutions. F or certain v alues of the parameters the equation for ϕ can b e integrated explicitly . In particular, we recov er the one-soliton solutions earlier found by means of the inv erse scattering approach [5]. W e also commen t on the o ccurence of more exotic tra v eling wa ves such as p eakons. A p eakon is a solution whose profile has a p eak at its crest, see Figure 3. The existence of p eak ed trav eling-wa v e solutions is a w ell-known prop erty of the Camassa-Holm equation [14]. Since (1.2) is related to the NLS equation by a procedure analogous to that whic h gives the Camassa-Holm equation from KdV, it is natural to inv estigate whether (1.2) also exhibits weak solu- tions of this kind. Our discussion in Section 4 suggests that (1.2) admits no p eak ed tra v eling wa v es which are w eak solutions in an y reasonable sense. 2. Physical deriv a tion Consider the propagation of an optical pulse in a monomo de fib er aligned in the z - direction with a frequency-dependent dielectric constan t. W e first follow the standard deriv ation of the NLS equation (see e.g. [1]), b efore we indicate ho w the additional higher-order effects particular to equation (1.2) arise. Maxwell’s equations lead to the basic equation (2.1) ∇ × ∇ × E = − 1 c 2 E tt − µ 0 P tt , where E ( r , t ) and P ( r , t ) are the electric field and the induced p olarization ev aluated at time t at the spatial p oin t r , resp ectiv ely , µ 0 is the v acuum p ermeabilit y , c is the sp eed of light in v acuum, and subscripts denote partial deriv atives. Assuming a lo cal medium resp onse and including only third-order nonlinear effects, the total p olarization P ( r , t ) = P L ( r , t ) + P N L ( r , t ) can be split in to a linear part P L =  0 Z t −∞ χ (1) ( t − t 0 ) E ( r , t 0 ) dt 0 , and a nonlinear part P N L =  0 Z t −∞ dt 1 Z t −∞ dt 2 Z t −∞ dt 3 χ (3) ( t − t 1 , t − t 2 , t − t 3 )[ E ( r , t 1 ) , E ( r , t 2 ) , E ( r , t 3 )] dt 0 , 4 JONA T AN LENELLS where  0 is the v acuum p ermittivity and χ ( j ) is the j th-order susceptibility . If we moreo v er assume that the nonlinear resp onse of the fib er is instan taneous, w e find (2.2) P N L ( r , t ) =  0 χ (3) [ E ( r , t ) , E ( r , t ) , E ( r , t )] , where χ (3) is a time-independent map linear in its three argumen ts. Since w e consider a quasi-monochromatic pulse whic h main tains its p olarization along the fib er length, a scalar approach is appropriate. If the pulse sp ectrum is cen tered at ω 0 , we write E ( r , t ) = ˆ x 2  E ( r , t ) e − iω 0 t + c.c.  , P L ( r , t ) = ˆ x 2  P L ( r , t ) e − iω 0 t + c.c.  , (2.3) P N L ( r , t ) = ˆ x 2  P N L ( r , t ) e − iω 0 t + c.c.  , where ˆ x is a unit vector p ointing in the direction of the p olarization, c.c. stands for the complex conjugate, and E , P L , P N L are complex-v alued slowly-v arying functions. Substituting (2.3) into (2.2) and ignoring terms of frequency 3 ω 0 , we deduce that P N L =  0  N L E , where  N L = 3 4 χ (3) xxxx | E ( r , t ) | 2 is the nonlinear contribution to the dielectric constan t resp onsible for the Kerr effect ( χ (3) xxxx denotes the xxxx -comp onen t of χ (3) ). T aking the F ourier transform of equation (2.1) while making the approximate assumption that  N L is a constant, w e obtain (2.4) ∇ 2 ˜ E ( r , ω − ω 0 ) +  ( ω ) k 2 0 ˜ E ( r , ω − ω 0 ) = 0 , where k 0 = ω /c , the F ourier transform ˜ E is defined by ˜ E ( r , ω − ω 0 ) = Z R E ( r , t ) e i ( ω − ω 0 ) t dt, and the frequency-dep endent dielectric constant  ( ω ) = 1 + ˜ χ (1) xx ( ω ) +  N L , con tains the F ourier transform ˜ χ (1) xx ( ω ) of the xx -comp onen t of χ (1) . W e seek a solution to (2.4) of the form ˜ E ( r , ω − ω 0 ) = F ( x, y ) ˜ A ( z , ω − ω 0 ) e iβ 0 z , where ˜ A ( z , ω ) is a slowly v arying function of z and β 0 is the w a v e num b er accounting for the fast oscillations. Substituting this into (2.4), we find F xx + F y y +   ( ω ) k 2 0 − ˜ β ( ω ) 2  F = 0 , (2.5) ˜ A z z + 2 iβ 0 ˜ A z +  ˜ β ( ω ) 2 − β 2 0  ˜ A = 0 , (2.6) where ˜ β ( ω ) is a function independent of r . A t this point we hav e deviated from the standard deriv ation in whic h the ˜ A z z term in equation (2.6) is ignored since A ( z , t ) is assumed to describe the slowly v arying env elope of the pulse. The deriv ation proceeds b y solving (2.6) using p erturbation theory and then apply- ing the in v erse F ourier transform in order to return to the time domain. The result is (see [1] for details in the case when ˜ A z z is ignored) (2.7) − i 2 β 0 A z z + A z + β 1 A t + iβ 2 2 A tt + α 2 A = iρ | A | 2 A, where the terms prop ortional to α and ρ include the effects of fib er loss and non- linearit y , resp ectiv ely . The co efficien ts β 0 , β 1 , β 2 are determined by the expansion of NONLINEAR PULSE PR OP AGA TION IN OPTICAL FIBERS 5 β ( ω ) = n ( ω ) ω /c , where n ( ω ) is the refractive index, around the carrier frequency ω 0 according to β ( ω ) = β 0 + β 1 ( ω − ω 0 ) + β 2 2 ( ω − ω 0 ) 2 + β 3 3! ( ω − ω 0 ) 3 + · · · . In the frame of reference trav eling with the pulse at the group velocity v g = 1 /β 1 , equation (2.7) b ecomes (2.8) iA z + 1 2 β 0 A z z − 1 β 0 v g A z T + γ A T T + iα 2 A = − ρ | A | 2 A, where γ = 1 2 β 0 v 2 g − β 2 2 and we ha v e introduced the new v ariable T b y T = t − β 1 z . As compared with the standard deriv ation, w e see that our retaining of ˜ A z z in equation (2.6) has giv en rise to t w o additional terms in (2.8) in v olving A z z and A z T . If these are ignored and the dissipation is set to zero (i.e. α = 0), equation (2.8) reduces to the NLS equation. The terms prop ortional to A z z and A z T are usually ignored due to the inequalities (see [15]) | A z z |  β 0 | A z | , | A z T |  β 0 v g | A z | , assumed to b e v alid if A is a slo wly v arying en v elop e. Ho w ever, in the (fem tosecond) regime of very short pulses, the pulse en v elop e A may con tain only a few optical cycles. Hence these inequalities are exp ected to b e less strong in this range and it motiv ates us to not throw a w ay the terms in v olving A z z and A z T . On the other hand, for v ery short pulses the appro ximation (2.2) also has to b e corrected in order to incorp orate higher-order nonlinear effects. A more careful analysis [8] taking additional effects in to account sho ws that the righ t-hand side of equation (2.8) should be replaced b y iβ 3 6 A T T T − ρA | A | 2 − is  A | A | 2  T − iτ A  | A | 2  T , where β 3 go v erns the effect of third-order dispersion, s accounts for the so-called self- steep ening effect, and τ is in general a complex parameter arising from the retarded nonlinear rep onse of the medium; in particular, the imaginary part of τ describes the retarded Raman effect in whic h a photon through scattering pass on part of its energy to a vibrational mo de of a molecule in the medium. The Raman effect is a nonlinear dissipative effect whic h causes a frequency downshift of the pulse [1, 8]. While our final equation iA z + 1 2 β 0 A z z − 1 β 0 v g A z T + γ A T T + iα 2 A − iβ 3 6 A T T T (2.9) = − ρA | A | 2 − is  A | A | 2  T − iτ A  | A | 2  T is rather complicated, there exist some sp ecial c hoices of the parameters for whic h the equation is in tegrable and hence exactly solv able by means of the inv erse scattering formalism. First of all w e need to set the linear and nonlinear dissipative terms to zero (i.e. α = 0 and Im τ = 0) in order to get an energy-conserving system. Then sev eral integrable reductions exist. • When the terms prop ortional to A z z , A z T , A T T T , and A  | A | 2  T are neglected, equation (2.9) reduces to (2.10) iA z + γ A T T = − ρA | A | 2 − is  A | A | 2  T . This equation is integrable and admits soliton solutions [16]. In fact, a gauge transformation conv erts it into the DNLS equation (1.3). 6 JONA T AN LENELLS • More generally , when the terms proportional to A z z and A z T are ignored, w e obtain (2.11) iA z + γ A T T − iβ 3 6 A T T T = − ρA | A | 2 − is  A | A | 2  T − iτ A  | A | 2  T . There are t wo known [17] integrable sub cases of this equation apart from (2.10); both o ccur when the parameters satisfy β 3 ρ = − 2 sγ . When this condition is fulfilled, a gauge transformation brings (2.11) to the form [18] (2.12) iA z − iβ 3 6 A T T T = − is | A | 2 A T − i ( s + τ ) A  | A | 2  T . If the ratios of the co efficients − β 3 6 , s , and s + τ in (2.12) are related as 1 : 6 : 0 w e obtain Hirota’s equation [19] A z + A T T T = − 6 | A | 2 A T , while if they are related as 1 : 6 : 3 w e find the Sasa-Satsuma equation [20] A z + A T T T = − 6 | A | 2 A T − 3 A  | A | 2  T . • When the terms prop ortional to A z z and A T T T are ignored and s + τ = 0, equation (2.9) b ecomes (2.13) iA z − 1 β 0 v g A z T + γ A T T = − ρ | A | 2  A + i s ρ A T  , whic h is equation (1.2) if 1 β 0 v g = s ρ . 1 W e conclude that these four exactly solv able mo dels arise as reductions of the full nonlinear equation (2.9) when different choices are made as to which higher-order effects to include. Although none of them can b e exp ected to exhibit all the structure tied in to (2.9) (e.g. they all ignore the effects of fiber loss), they do pro vide important first approximations to the full theory . In this sense equation (1.2) can be view ed as a mo del for ligh t pulses in single-mode optical fib ers. 3. Rela tion to DNLS It w as found in [5] that the x -part of the Lax pair of (1.2) is simply related to the x -part of the Lax pair of the deriv ativ e nonlinear Schr¨ odinger equation (1.3). More precisely , the t w o x -parts are iden tical if u x is iden tified with q , where u is the solution of (1.2) and q is the solution of (1.3). Although the corresp onding t -parts are v ery differen t, this suggests a deep relationship b et w een (1.2) and (1.3). Since all mem b ers of the hierarc hy asso ciated with an integrable equation admit the same x -part, it is reasonable to expect the equation describing the ev olution of u x according to (1.2) to b e related to a member of the DNLS hierarch y . W e will show that a simple change of v ariables indeed transforms (1.2) in to the first negativ e mem b er of this hierarc h y . 3.1. Gauge transformation. Our first step is to transform equation (1.2) b y means of a gauge transformation. Replacing x by − x if necessary , we can assume that the parameters γ and ν in (1.2) hav e the same sign. Then, letting a = γ /ν > 0 and b = 1 /ν , the gauge tranformation u → r a | ρ | be i ( bx +2 abt ) u, 1 The v ariables A , z , and T in (2.13) are iden tified with u , t , and x in (1.2), resp ectively . The parameter ν is identified with 1 β 0 v g = s ρ . NONLINEAR PULSE PR OP AGA TION IN OPTICAL FIBERS 7 transforms (1.2) into u tx − au xx = ab 2 ( − u + iσ | u | 2 u x ) , σ = sgn ρ = ± 1 . In tro ducing new v ariables by ξ = x + at, τ = − ab 2 t, w e arrive at the equation (3.1) u τ ξ = u − iσ | u | 2 u ξ , σ = ± 1 , whic h is exactly (1.4) up to a relab eling of τ and ξ . Putting the v arious transforma- tions together we arriv e at the follo wing result. Prop osition 3.1. Fix any nonzer o values of the p ar ameters ν, γ , ρ ∈ R . Then u ( x, t ) satisfies e quation (1.2) if and only if u g ( ξ , τ ) satisfies e quation (3.1) with σ = sgn( ν γ ρ ) wher e u and u g ar e r elate d as u ( x, t ) = r sgn( ν γ ρ ) γ ν 3 ρ e i “ x ν + 2 γ t ν 2 ” u g  x + γ ν t, − γ ν 3 t  . 3.2. Bi-Hamiltonian structure. Equations (1.3) and (1.4) are equiv alent to the systems (3.2)  q r  t =  iq xx − ( q 2 r ) x − ir xx − ( q r 2 ) x  and (3.3)  u x v x  t =  u − iuv u x v + iuv v x  , resp ectiv ely , when r = σ ¯ q and v = σ ¯ u . The op erators J 1 =  0 ∂ x ∂ x 0  , J 2 =  − q ∂ − 1 x q i + q ∂ − 1 x r − i + r ∂ − 1 x q − r ∂ − 1 x r  , form a compatible pair of Hamiltonian op erators cf. [21]. Starting with the Hamil- tonian H 0 = − Z q r dx, the operators J 1 and J 2 generate a hierarc h y of bi-Hamiltonian equations according to 2  q r  t = K n = J 1 grad H n − 1 = J 2 grad H n , n ∈ Z , where the infinite sequence of conserv ation la ws { H n } n ∈ Z are constructed recursively from the relation J 1 grad H n = J 2 grad H n +1 , n ∈ Z . The first few members of this hierarch y are presented in Figure 1, where we hav e written u and v for ∂ − 1 x q and ∂ − 1 x r , respectively . W e immediately recognize equation 2 The gradien t of a functional F [ q , r ] is defined by grad F = „ δF δq δF δr « , whenev er there exist functions δF δq and δF δr suc h that, for any smo oth functions ϕ 1 and ϕ 2 , d d F [ q + ϕ 1 , r + ϕ 2 ] ˛ ˛ ˛ ˛  =0 = Z „ δ F δ q ϕ 1 + δ F δ r ϕ 2 « dx. 8 JONA T AN LENELLS grad R  q r xx − 1 2 q 3 r 3 − 3 i 2 q 2 r r x  dx = grad H 2 J 2 s s g g g g g g g g g g g g g g g g g g g g  q r  t =  iq xx − ( q 2 r ) x − ir xx − ( q r 2 ) x  grad R  − 1 2 q 2 r 2 − iq r x  dx = grad H 1 J 1 k k W W W W W W W W W W W W W W W W W W W W J 2 s s g g g g g g g g g g g g g g g g g g g g g g g g g g  q r  t = −  q x r x  grad R ( − q r ) dx = grad H 0 J 1 k k W W W W W W W W W W W W W W W W W W W W W W W W W W J 2 s s g g g g g g g g g g g g g g g g g g g g g g g g g g  q r  t = i  − q r  grad R iv u x dx = grad H − 1 J 1 k k W W W W W W W W W W W W W W W W W W W W W W W W W W J 2 s s g g g g g g g g g g g g g g g g g g g g g g g  u x v x  t =  u − iuv u x v + iuv v x  grad R  − uv + i 2 uv 2 u x  dx = grad H − 2 J 1 k k W W W W W W W W W W W W W W W W W W W W W W Figure 1. R e cursion scheme for the op er ators J 1 and J 2 . The notation u = ∂ − 1 x q and v = ∂ − 1 x r has b e en use d when c onvenient. (3.2) as the second member and equation (3.3) as the first negative mem b er of this hierarc h y . Remark 3.2. Since the in v erse of J 2 is given explicitly b y J − 1 2 =  r ∂ − 1 x r i + r ∂ − 1 x q − i + q ∂ − 1 x r q ∂ − 1 x q  , it is straightforw ard to implement the recursive scheme for construction of the H n ’s on a computer and thus obtain explicit formulas for as many conserv ed quan tities as one wishes. The H n ’s are lo cal in q and r for n ≥ 0, while they in v olve the nonlo cal op erator ∂ − 1 x for n < 0. In b oth cases, the expressions grow rapidly in size, e.g. H 3 NONLINEAR PULSE PR OP AGA TION IN OPTICAL FIBERS 9 and H 4 are given b y H 3 = Z  − 5 8 q 4 r 4 − 5 2 iq 3 r x r 2 + 1 2 q q xx r 2 + q q x r x r + 2 q 2 r xx r + 3 2 q 2 r 2 x + iq r xxx  dx, H 4 = Z  − 7 8 q 5 r 5 + 5 6 q q 2 x r 3 − 35 8 iq 4 r x r 3 + 5 3 q 2 q xx r 3 + 5 q 2 q x r x r 2 + 25 6 q 3 r xx r 2 + 35 6 q 3 r 2 x r + 5 2 iq r x q xx r + 5 2 iq q x r xx r + 5 2 iq 2 r xxx r + 5 2 iq q x r 2 x + 5 iq 2 r x r xx − q r xxxx  dx. 4. Tra veling w a ves In this section w e analyze the trav eling-w a ve solutions of equation (1.2) taken in the simpler, but equiv alen t, form (1.4). Letting ( t, x ) → ( σ t, σ x ), w e may assume that σ = 1 in (1.4). Th us consider the equation (4.1) u tx = u − i | u | 2 u x . Substituting u ( x, t ) = W ( x, t ) e iχ ( x,t ) , with real-v alued functions W and χ in to (4.1), we obtain the coupled equations ( − χ x W 3 − ( χ t χ x + 1) W + W tx = 0 , W x W 2 + χ tx W + W t χ x + χ t W x = 0 . (4.2) W e seek a solution of the form (4.3) χ ( x, t ) = k x − Ω t + θ ( y ) , W ( x, t ) = ϕ ( y ) , where k , Ω, c are real parameters and y = x − ct . Equations (4.2) and (4.3) yield ( − ( k + θ y ) ϕ 3 − (1 − ( k + θ y ) (Ω + cθ y )) ϕ − cϕ y y = 0 − ϕ y  − ϕ 2 + ck + Ω + 2 cθ y  − cϕθ y y = 0 (4.4) Multiply the second of these equations by ϕ and integrate the resulting equation to find ϕ 4 4 − 1 2 ( ck + Ω + 2 cθ y ) ϕ 2 + A = 0 , for some constant of integration A . Solving for θ y , we obtain (4.5) θ y = ϕ 4 − 2 ck ϕ 2 − 2Ω ϕ 2 + 4 A 4 cϕ 2 . W e use this equation to eliminate θ y from the first equation in (4.4). In tegration of the resulting equation m ultiplied by ϕ y yields (4.6) ϕ 2 y = − 1 16 c 2 V ( ϕ ) , 10 JONA T AN LENELLS  2  4  6 2 4 6 x  0.5 0.5 Re u  2  4  6 2 4 6 x  0.5 0.5 Im u  2  4  6 2 4 6 x  0.5 0.5  u  Figure 2. The r e al p art, imaginary p art, and the absolute value at t = 0 of the one-soliton solution u ( x, t ) given by e quation (4.7). The gr aphs c orr esp ond to the p ar ameter-values ∆ = 1 , γ = 1 / 2 , and Σ 0 = Θ 0 = 0 . where V ( ϕ ) = ϕ 8 + c 3 ϕ 6 + c 2 ϕ 4 + c 1 ϕ 2 + c 0 ϕ 2 , c 3 = 4 ck − 4Ω , c 2 = 4  c 2 k 2 + Ω 2 + 2 A + c (4 − 2 k Ω)  , c 1 = − 32 B c, c 0 = 16 A 2 , and B is another integration constant. Solutions of (4.6) can only exist in regions where V ( ϕ ) ≤ 0. Since c 0 > 0 unless A = 0, w e infer that solitary w a v es (i.e. wa v es suc h that ϕ deca ys to zero as y → ±∞ ) can exist only when A = 0. A classification of the tra v eling wa ves can b e obtained b y analyzing the distribution of the zeros of V as the parameters Ω , c, k , A, B v ary . Rather than completing this program in detail, w e simply indicate ho w the one-soliton solutions arise and discuss the (non-)existence of p eaked tra v eling wa ves. 4.1. One-solitons. Equation (4.1) admits the one-soliton solutions [5] (4.7) u s ( x, t ) = − 2 ie i Σ+Θ+2 iγ sin γ ∆( e 2Θ+ iγ + 1) , where Σ =  t − 4∆ 4 x  cos γ 2∆ 2 + Σ 0 , (4.8) Θ =  t + 4 x ∆ 4  sin γ 2∆ 2 + Θ 0 , (4.9) NONLINEAR PULSE PR OP AGA TION IN OPTICAL FIBERS 11 and γ ∈ (0 , π ) , ∆ > 0 , Σ 0 ∈ R , Θ 0 ∈ R , are four parameters, see Figure 2. In order to relate this family of one-solitons to the ab ov e description of tra veling w a ves, we need to find parameters c s , k s , Ω s , and functions θ s ( y ) , ϕ s ( y ) such that (4.10) u s ( x, t ) = ϕ s ( x − c s t ) e i ( k s x − Ω s t + θ s ( x − c s t )) . The amplitude of u s satisfies | u s | 2 = 2 sin 2 γ ∆ 2 (cos γ + cosh 2Θ) . This yields (4.11) ϕ s ( y ) = s 2 sin 2 γ ∆ 2 (cos γ + cosh (4∆ 2 y sin γ + 2Θ 0 )) , c s = − 1 4∆ 4 . It can b e verified that the function ϕ s ( y ) given by (4.11) satisfies the ODE (4.6) pro vided that A = B = 0 and (4.12) k s = 0 , Ω s = − cos γ ∆ 2 . Substitution of expression (4.11) for ϕ s in to (4.5) giv es after integration θ s ( y ) = − arctan  tan  γ 2  tanh  2∆ 2 y sin γ + Θ 0   (4.13) − cot( γ )  2∆ 2 y sin γ + Θ 0  + θ 0 s , where θ 0 s is a constan t of in tegration. W e can determine θ 0 s b y ev aluating (4.10) at x = t = 0. This yields (4.14) e iθ 0 s = − i √ 2 e i ( 2 γ − i Θ 0 +Σ 0 +arctan ( tan ( γ 2 ) tanh Θ 0 ) +Θ 0 cot γ ) p cos γ + cosh(2Θ 0 ) 1 + e iγ +2Θ 0 . In summary , the one-soliton u s can b e written in the form (4.10) where c s , k s , Ω s , θ s ( y ) , ϕ s ( y ) are giv en b y equations (4.11)-(4.14). This shows how the one-soliton solutions are related to the ab ov e tra v eling-w av e analysis. 4.2. P eak ons. The Camassa-Holm equation is w ell-kno wn for its p eak ons, which are solitons with a peak at their crest [14] (see Figure 3). Since equation (1.2) is related to the NLS equation in a similar wa y that the Camassa-Holm equation is related to KdV, one may w onder if (1.2) also admits some kind of p eaked solutions. 3 Let us first make some general commen ts on equations of the form (4.6) and the o ccurence of peakons. The ordinary differential equation (4.6) can b e analyzed qual- itativ ely by regarding ϕ ( y ) as describing the motion of a ball rolling in a p otential w ell V ( ϕ ) with y representing time. Using this analogy , p eak ed solitary w av es arise as follo ws. Consider a p oten tial V ( ϕ ) with a qualitative shap e as in Figure 4. Let the ball at time y = −∞ b e lo cated at the crest of the hill at the p oin t A . Give it a gen tle push so that it starts rolling down the v alley . If nothing interv enes, it will clim b the opp osite side of the v alley un til it reac hes the p oin t B where it turns 3 More precisely , we sa y that a contin uous real-v alued function f ( x ) has a peak at x 0 if f is smooth lo cally on either side of x 0 and the left and right deriv atives of f at x 0 are b oth finite but not equal. A p eak ed solution u ( x, t ) is a solution such that, for eac h fixed time t , u ( · , t ) has at least one p eak. In the case of a complex-v alued solution u ( x, t ), one could allow peaks in either the real or the imaginary part of u ( x, t ), or in b oth. 12 JONA T AN LENELLS 0 x Figure 3. A p e akon. The discussion in Subse ction 4.2 suggests that e quation (1.2) admits no p e ake d solutions. Figure 4. An example of a p otential wel l V ( ϕ ) . around; it crosses the v alley again and has just enough energy to climb all the wa y bac k up to the starting p oin t A as y → ∞ . This motion corresp onds to a smo oth solitary w av e solution. How ev er, if we in tercept the solution b efore it reac hes p oin t B at some p oint C at which we let it (after a p erfectly elastic collision) reb ound, then it will climb up the first hill again and approach its starting p osition A as y → ∞ . The effect of the b ounce is to create a peakon singularit y at whic h the deriv ativ e ϕ y jumps, ϕ y → − ϕ y . Clearly , for most soliton equations there are plen t y of wa ys to in tercept solutions in this manner and thus obtain p eaked solitons (see [22] for an example of this in the con text of fiber optics). How ev er, ev en though the peaked tra v eling w a v es so ob- tained satisfy the original PDE wherever they are smooth, the singularities in general in tro duce additional δ -functions so that the PDE is not satisfied in a reasonable weak sense. F or most equations this rules out the existence of p eaked solitons. On the other hand, if w e consider the Camassa-Holm equation (4.15) u t − u txx + 3 uu x = 2 u x u xx + uu xxx , the followin g happ ens. F or a trav eling w a v e u ( x, t ) = ϕ ( x − ct ), in tegration of (4.15) yields (4.16) − cϕ + ( c − ϕ ) ϕ xx + 3 2 ϕ 2 = 1 2 ϕ 2 x + a 2 , for some constan t of integration a ∈ R . In going from (4.15) to (4.16), the u txx and uu xxx terms hav e com bined to give the term ( c − ϕ ) ϕ xx . Moreo v er, ( c − ϕ ) ϕ xx is the only term in (4.16) whic h in volv es a second deriv ativ e of ϕ . At a p oin t where ϕ has a p eak, ϕ xx is prop ortional to a δ -function. Hence, the singularity-induced δ -function in (4.16) at a peak appears m ultiplied by c − ϕ , where c is the sp eed of the NONLINEAR PULSE PR OP AGA TION IN OPTICAL FIBERS 13 w a ve. One could therefore hop e for a peak to b e allo wed at a p oint x 0 pro vided that c − ϕ ( x 0 ) = 0, since this factor then kills the δ -function. This is indeed what happ ens: a peak in a trav eling wa v e of the Camassa-Holm equation is admissible exactly when the heigh t of the peak equals the sp eed of the wa v e, i.e. when ϕ = c at the crest cf. [23]. On the con trary , we will no w argue that for equation (1.2) the δ -functions produced b y the x -deriv atives of a p eak on candidate stand alone and cannot b e cancelled by other factors. F or simplicit y we consider the equiv alent, but simpler, equation (4.1); the argumen t for (1.2) is analogous. When substituting a tra v eling-w a ve solution of the form (1.5) in to equation (4.1), the terms inv olving a second deriv ativ e of ϕ or θ can b e read off from equation (4.4). They are − cϕ y y in the equation for the real part, − cϕθ y y in the equation for the imaginary part. F or the solution to hav e a p eak at y 0 at least one of the functions ϕ y y or θ y y m ust in v olve a δ -function at y 0 . The only wa y for the real and the imaginary parts to v anish simultaneously is therefore that either (i) c = 0 or (ii) only the function θ has a peak at y 0 and ϕ ( y 0 ) = ϕ y y ( y 0 ) = 0. The follo wing observ ations indicate that neither of these t w o options is viable. Regarding (i) we note that if c = 0 then the second equation in (4.4) b ecomes − ϕ y ( − ϕ 2 + Ω) = 0 , y ∈ R . By the assumption that the amplitude of a p eakon is a contin uous function, this sho ws that the amplitude ϕ is in fact constan t. It then follo ws from the first equation in (4.4) that θ y is also a constan t function given b y θ y = k ϕ 2 − k Ω + 1 Ω − ϕ 2 . Th us, no peakons arise when c = 0. Regarding (ii) w e note that a v anishing amplitude ϕ ( y 0 ) = 0 at y 0 implies that the phase θ ( y 0 ) is undetermined a priori. Of course θ may contain p eaks (or jumps for that matter) in an y interv als where ϕ = 0, but these p eaks are unobserv able and are just a consequence of the breakdo wn of the p olar co ordinate system at the origin. As soon as ϕ becomes nonzero the v alue of θ y is fixed by the equations in (4.4) and cannot jump. Hence, neither of the options (i) or (ii) leads to existence of peakons. The ab ov e discussion indicates that (4.1) admits no p eaked solutions within the class of trav eling w a ves of the form (1.5). Although one could imagine that there exist p eak ed solutions of a more general t yp e, similar argumen ts as ab o ve applied directly to equation (4.1) make this unlikely (since u tx is the only term in the equation inv olving t w o deriv ativ es, there is no other term to balance the δ -function generated b y u tx at the p eak). In conclusion, it app ears that equation (1.2) admits no peakons which are w eak solutions in a reasonable sense. References [1] G. P . Agra w al, Nonline ar Fib er Optics , fourth edition, Academic Press, 2007. [2] V. E. Zakharo v and A. B. Shabat, Exact theory of t w o-dimensional self-focusing and one- dimensional self-mo dulation of wa v es in nonlinear media, Soviet Phys. JETP 34 (1972), 62–69. [3] A. S. F ok as, On a class of physically imp ortan t integrable equations, Phys. D 87 (1995), 145–150. 14 JONA T AN LENELLS [4] B. F uc hssteiner and A. S. F ok as, Symplectic structures, their B¨ ac klund transformation and hereditary symmetries, Phys. D 4 (1981), 47–66. [5] J. Lenells and A. S. F ok as, On a no vel integrable generalization of the nonlinear Scr¨ odinger equation, submitted. [6] F. Calogero, Wh y are certain nonlinear PDEs b oth widely applicable and in tegrable?, in: What is integrabilit y? (V.E.Zakharo v, editor), Springer, 1990, 1–62. [7] Y. Kodama, Optical solitons in a monomo de fib er, J. Stat. Phys. 39 (1985), 597–614. [8] Y. Ko dama and A. Hasewaga, Nonlinear pulse propagation in a monomo de dielectric guide, IEEE J. Quantum Ele ctr on. 23 (1987), 510–524. [9] D. Anderson and M. Lisak, Nonlinear asymmetric self-phase mo dulation and self- steep ening of pulses in long optical wa v eguides, Phys. R ev. A 27 (1983), 1393–1398. [10] N. Tzoar and M. Jain, Self-phase modulation in long-geometry optical w av eguides, Phys. R ev. A 23 (1981), 1266–1270. [11] E. Mjolh us, On the mo dulational instability of hydromagnetic wa v es parallel to the magnetic field, J. Plasma Phys. 16 (1976), 321–334. [12] D. J. Kaup and A. C. New ell, An exact solution for a deriv ative nonlinear Schr¨ odinger equation, J. Math. Phys. 19 (1978), 789–801. [13] H. P . McKean, The Liouville corresp ondence b etw een the Korteweg-de V ries and the Camassa-Holm hierarc hies, Comm. Pur e Appl. Math. 56 (2003), 998–1015. [14] R. Camassa and D. D. Holm, An integrable shallo w w ater equation with p eak ed solitons, Phys. R ev. L ett. 71 (1993), 1661–1664. [15] J. Hermmann and B. Wilhelmi, L asers for ultr ashort light pulses , North-Holland, 1987. [16] H. H. Chen, Y. C. Lee and C. S. Liu, In tegrabilit y of nonlinear Hamiltonian systems b y in verse scattering method, Phys. Scr. 20 (1979), 490–492. [17] S. Y u. Sako vic h, In tegrabilit y of the higher-order nonlinear Sc hr¨ odinger equation revis- ited, arXiv:solv-int/9906012v1. [18] C. Gilson, J. Hietarin ta, J. Nimmo, and Y. Oh ta, Sasa-Satsuma higher order nonlinear Sc hr¨ odinger equation and its bilinearization and multi-soliton solutions, Phys. R ev. E 68 (2003), 016614. [19] R. Hirota, Exact env elope-soliton solutions of a nonlinear w av e equation, J. Math. Phys. 14 (1973), 805–809. [20] N. Sasa and J. Satsuma, New-t ype soliton solution for a higher-order nonlinear Sc hrdinger equation, J. Phys. So c. Jpn. 60 (1991), 409–417. [21] W. Ma and R. Zhou, On inv erse recursion op erator and tri-Hamiltonian form ulation for a Kaup-Newell system of DNLS equations, J. Phys. A 32 L239–L242. [22] L. Cheng, The cusp solitons in optical fibres, J. Phys. A , 29 (1996), 141–142. [23] J. Lenells, T rav eling wa v e solutions of the Camassa-Holm equation, J. Diff. Eq. , 217 (2005), 393–430. J.L.: Dep ar tment of Applied Ma thema tics and Theoretical Physics, University of Cambridge, Cambridge CB3 0W A, UK E-mail addr ess : j.lenells@damtp.cam.ac.uk