Soliton dynamics in an extended nonlinear Schrödinger equation with a spatial counterpart of the stimulated Raman scattering

Transcription

1 Soliton ynamics in an extene nonlinear Schröinger equation with a spatial counterpart of the stimulate Raman scattering E.M. Gromov 1a an B.A. Malome b a National Research University Higher School of Economics, 5/1 Bolshaja Pecherskaja Ulitsa, Nizhny Novgoro 63155, Russia b Department of Physical Electronics, Faculty of Engineering, Tel Aviv University, Tel Aviv, 69978, Israel Dynamics of solitons is consiere in the framework of the extene nonlinear Schröinger equation (NLSE), which is erive from a system of Zakharov's type for the interaction between high- an low-frequency (HF an LF) waves, in which the LF fiel is subject to iffusive amping. The moel may apply to the propagation of HF waves in plasmas. The resulting NLSE inclues a pseuostimulate-raman-scattering (PSRS) term, i.e., a spatial-omain counterpart of the SRS term which is well known as an ingreient of the temporal-omain NLSE in optics. Also inclue is inhomogeneity of the spatial secon-orer iffraction (SOD). It is shown that the wavenumber ownshift of solitons, cause by the PSRS, may be compensate by an upshift provie by the SOD whose coefficient is a linear function of the coorinate. An analytical solution for solitons is obtaine in an approximate form. Analytical an numerical results agree well, incluing the preicte balance between the PSRS an the linearly inhomogeneous SOD. 1. Introuction The great interest to the ynamics of solitons is motivate by their ability to travel large istances, preserving a permanent shape. Soliton solutions are generate by various nonlinear moels ealing with the propagation of intense wave fiels in ispersive meia: optical pulses in fibers, electromagnetic waves in plasma, surface waves on eep water, etc. (Agrawal 1; Kivshar & Agrawal 3; Dauxois & Peyrar 6; Dickey 5; Infel & Rowlans ; Malome 6; Yang 1). Dynamics of long high-frequency (HF) wave packets is escribe by the secon-orer nonlinear ispersive wave theory. The funamental equation of the theory is the nonlinear Schröinger equation (NLSE) (Zakharov & Shabat 197; Hasegava & Tappert 1973), which inclues the secon-orer ispersion (SOD) an cubic nonlinearity (self-phase moulation). Soliton solutions in this case arise as a result of the balance between the ispersive stretch an nonlinear compression of the wave packet. The ynamics of short HF wave packets is escribe by the thir-orer nonlinear ispersive wave theory (Agrawal 1), which takes into account the nonlinear ispersion (self-steeping) (Oliviera & Moura 1998), stimulate Raman scattering (SRS) in optical meia (Mitschke & Mollenauer 1986; Goron 1986; Koama 1985), an thir-orer ispersion (TOD). Accoringly, the basic equation of the theory is the thir-orer NLSE (Koama 1985; Koama & Hasegava 1987; Zaspel 1999; Karpman 4; Hong & Lu 9). Soliton solutions in the framework of the thir-orer NLSE without the SRS term were investigate in Refs. (Gromov & Talanov 1996, ; Gromov, Piskunova & Tyutin 1999; Obregon & Stepanyants 1998; Scalora at al 5; Wen at al 6; Marklun, Shukla & Stenflo 6; Tsitsas at al 9). Such solitons are supporte by the balance between the TOD an nonlinear ispersion. In (Kivshar 199; Kivshar & Malome 1993) stationary 1 aress for corresponence: egromov@hse.ru 1

2 kink moes were foun as solutions of the thir-orer NLSE without the TOD term. This solution exists ue to the balance between the nonlinear ispersion an SRS. For localize nonlinear wave packets (solitons), the SRS gives rise to the ownshift of the soliton spectrum (Koama 1985; Goron 1986; Mitschke & Mollenauer 1986) an eventually to estabilization of the solitons. The use of the balance between the SRS an the slope of the gain for the stabilization of solitons in long telecom links was propose by Malome & Tasgal (1998). The compensation of the SRS by emission of linear raiation fiels from the soliton's core was consiere in by Biancalama, Skrybin & Yulin (4). The compensation of the SRS in inhomogeneous meia was consiere in other settings too: perioic SOD (Essiambre & Agraval 1997a, 1997b), sliing zero-ispersion point (Anrianov, Muraviev, Kim & Sysoliatin 7), an ispersion-ecreasing fibers (Chernikov, Dianov, Richarson & Payne 1993). In this work the ynamics of HF wave packets is consiere in ispersive nonlinear meia, taking into account the interaction of the HF fiel with low-frequency (LF) waves, which are subject to iffusive amping. In the thir-orer approximation of the ispersion-wave theory, the original Zakharov-type system of equations for the HF an LF fiels (the LF equation inclues the iffusion/viscosity term), is reuce to an extene NLSE, which features a spatial counterpart of the SRS term, that we call a pseuo-srs (PSRS) one. The moel may be realize for the propagation of nonlinear waves in plasmas. The PSRS leas to the self wavenumber ownshift, similar to what is well known in the temporal omain (Agrawal 1; Goron 1986; Koama 1985; Mitschke & Mollenauer 1986). On the other han, the SOD term with a spatially ecreasing coefficient leas to an increase of the soliton's wavenumber. The balance between the PSRS an the ecreasing SOD leas to stabilization of the soliton's wavenumber spectrum. An analytical soliton solution is foun in an approximate form. The NPSE is erive in Section an analytical results are presente in Section 3. Numerical finings, an their comparison to the analytical preictions, are summarize in Section 4. The paper is conclue by Section 5.. The basic equation an integrals relations We consier the evolution of slowly varying envelope U ( ξ,t) of the intense HF wave fiel in the nonlinear meium with inhomogeneous SOD, taking into account the interaction with LF n ξ,t (such as the refractive inex in optics), which suffers variations of the meium's parameter the action of effective iffusion. The uniirectional propagation of the fiels along coorinate ξ is escribe by the system of the Zakharov's type (Zakharov 1971, 1974): U U i + q = t n n n + µ t ( ξ ) nu U, (1) =, () where µ is the iffusion coefficient. In particular, this system may escribe intense electromagnetic or Langmuir waves in plasmas, taking into account the scattering on ion-acoustic waves, which are subject to the viscous amping. The secon-orer approximation of the ispersion-wave theory correspons to replacing Eq. () by aiabatic relation packet obeys the NSLE: i U / t + ( q U / )/ + αu U = k n = U, hence envelope U of the HF wave ξ, where α = 1/. In the thir-orer approximation of the theory (for short HF wave packets, with k << µ, where an are the spatial extension an characteristic wave number of the wave packet), Eq. () may

3 approximate by the nonlinear response of the meium, n U ( U ) the following extene NLSE for the HF amplitue: = µ / ξ, which leas to U U ( U ) i + q( ξ ) + αu U + µ U =. (3) t The last term in Eq. (3) represents the above-mentione PSRS effect in the spatial omain. Equation (3) with zero bounary conitions at infinity, U, gives rise to the following ξ ± integral relations for fiel moments, which will be use below: the conservation of the wave action, N (a well-known property of the SRS term): + N U t t =, (4) the rate of change of the wave-fiel momentum: ( U ) q U KU µ t = ξ, (5) ξ the rate of change of the integrate square absolute value of the graient of the wave fiel: U U 1 U = µ K + α K K 3 K U ξ, (6) U q U t the rate of change of the square graient of the wave-fiel intensity: t t ( ) ( U U ) ( qk U ) ξ = + ξ the equation of motion for the center-of-mass coorinate, + +, (7) + 1 ξ N ξ U : ξ U = qk U, (8) where the complex fiel is represente as U U exp( iφ ) wavenumber. 3. Analytical results, an K φ / ξ is the local For analytical consieration of the wave-packet ynamics, we assume that the scales of the inhomogeneity of both the SOD term an local wavenumber K are much larger than the size of the wave-packet envelope, hence the spatial variation of the wavenumber may be locally approximate by the linear function of the coorinate, K( ξ, t) K( ξ, t) ( K / ξ) ξ ( ξ ξ) +. Then we obtain from the imaginary part of Eq. (3) uner conition ( U / ) = (which means that the peak of ξ the soliton's amplitue is locate at its center): K ξ ξ U 1 q = + K. (9) ξ qu t q ξ 3

4 Further, replacing K(, t ) ξ for soliton-like wave packets by K(, t) k( t) ξ, the system of Eqs. (4)- (7) can be cast in the form of evolution equations for parameters of the wave packet: k L = µ l q ( ξ ) z, t N (1) z L 3 = µ kl 3 kq ( ξ) z + k q ( ξ), t N (11) l = 3 kq ( ξ )l. t (1) q ξ = q / is the graient of the SOD coefficient at the center of the packet, l = L / L, Here ξ z = Z / N, an + + / ξ ξ, L ( U / ) Z Z U ξ, along with the conserve wave action N, are integral characteristics of the wave packet, Z =, L = L being their initial values. An equilibrium state (fixe point) of Eqs. (1)-(1) correspons to conitions k =, µ L = q ( ξ ) Z. (13) To analyze the ynamics of the wave packet with non-equilibrium parameters, we assume that the SOD coefficient is a linear function of the coorinate, q ( ξ ) = q + q ξ. (14) Then, substitutions τ tq an p µ L / ( q N ) reuce Eqs. (1)-(1) to k z l = z pl, = ( 3z pl k ) k, = 3kl. (15) τ τ τ The first integral of Eqs. (15) is k / z 3 3 ( 4k / z ) l ( 1 k / z ) l + λ = + λ l, (16) where k =, λ p / z, z Z / N. In Fig. 1, this relation between variables plotte for k k = an ifferent values of λ. k / z an l is Figure 1. Relation (16) for the soliton variables in the plane of ( k z, l) values of λ. 4 / for k = an ifferent We now look for stationary solutions to Eq. (3), where the SOD with spatial profile (14) is U ξ, t = ψ ξ exp iωt : aopte, in the form of a stationary wave profile, ψ ψ ψ = 3 + q ξ + q + αψ Ωψ + µψ q. (17)

5 Next, with regar to the unerlying assumption that the soliton's with is much smaller than the scale of the spatial inhomogeneity for the SOD than for the packet's envelope, a solution to Eq. (17) is foun in the form of ψ = ψ + ψ1, where ψ 1 is a small correction to ψ. In this approximation, we obtain ψ 3 q + αψ Ωψ =, (18) 3 ψ ( ψ ) αψ ψ q Ω = ξ µ q ψ1 ψ q. (19) 3 Equation (18) gives rise to the classical soliton solution, ψ A ( ξ ) an Ω αa /. Then substitutions = sech /, where q / α / A η = ξ / an Ψ = ψ q ( A q ) 1 / η cast Eq. (19) in the form of Ψ 6 η η 5 µ sinhη sinhη η Ψ = + +, () η η coshη 4 µ 4 cosh cosh * cosh η cosh η where the equilibrium value of the PSRS coefficient is µ 5q η α / q / A. (1) Uner conition Ψ( ) = * 8, an exact solution to Eq. () can be foun, η µ 1 µ Ψ( η) = Ψ η tanhη + ( tanhη) ln( coshη) sechη + 1 ( tanh η) sinhη 4 4µ * 1 µ, () * cf. a similar solution reporte by Blit & Malome (1). At µ = µ *, it satisfies bounary conitions Ψ( η ± ). This spatially antisymmetric solution, which is isplaye in Fig. for µ = µ * an ifferent values of Ψ, exists ue to the balance between the PSRS term an linearly ecreasing SOD. At µ, solution () iverges at the spatial infinity. µ * Figure. Solution () for ( ) = Ψ, µ = µ *, an ifferent values of Ψ. 4. Numerical results We numerically solve Eq. (3) with units fixe by setting α 1, ifferent values of the PSRS coefficient µ, linear SOD profile q ( ξ ) = 1 ξ / 1 (cf. Eq. (14)), an initial conitions corresponing to the soliton, U ( ξ, t = ) = sechξ. In this case, Eq. () preicts the equilibrium value of the PSRS coefficient, µ * = 1/16, for this initial pulse. As shown in Fig. 3, in irect simulations of Eq. (3) with 5

6 µ = 1/16 the input evolves into a stationary localize pulse with zero wavenumber, which is close to the above analytical solution, see Eq. (), with q = α = A = 1, q = 1/ 1, µ = µ * : 1 U = 1+ ( tanhξ ) ln( coshξ ) ξ tanhξ ) sechξ. (3) 4 In Fig. 3, the profile of the soliton solution (3) is shown by the otte curve. Fig. 3. The soli curve is the result of the numerical solution of Eq. (3) for the soliton's envelope, U( ξ ), obtaine in this stationary form in the time interval 1 < t 3, for q ( ξ ) = 1 ξ / 1 an µ = 1/ 16. The otte curve is the analytical solution (3). At values of the PSRS coefficient ifferent from µ *, the simulations prouce nonstationary solitons, with variable amplitue an wavenumber, see an example for µ = 5/18 (5/8) µ * in Figs. 4,5 Figure 4. The numerically simulation evolution of the wave-packet envelope versus ξ, t for µ = 5 / 18. 6

7 In Fig. 5, numerical results prouce, as functions of time, by the simulations for the local wavenumber at the maximum of the wave-packet's shape, are compare with the analytical counterparts obtaine from Eq. (1) for ifferent values of µ. Close agreement between the analytical an numerical results is observe by the figure, both for µ = µ *, when both the numerically an analytically foun wavenumbers remain equal to zero, an for nonstationary pulses µ µ at *. A similar picture is observe at other values of the parameters. 5. Conclusion Figure 5. Numerical an analytical results (soli an ashe curves) for the local wavenumber at the point of the maximum of the wave-packet envelope versus t for the SOD profile q ( ξ ) = 1 ξ / 1 an ifferent values of the PSRS coefficient µ. We have propose a moel which realizes the extene NLSE with the spatial-omain counterpart of the SRS term (the PSRS, i.e., pseuo-srs one). The equation is erive from the system of the Zakharov's type for electromagnetic an Langmuir waves in plasmas, in which the LF fiel is subject to the iffusive amping. We have stuie the soliton ynamics is the framework of the extene NLSE, which also inclues the smooth spatial variation of the SOD (secon-orer ispersion) coefficient. The analytical preictions were prouce by integral relations for the fiel moments, an numerical results were generate by systematic simulations of the pulse evolution in the framework of the extene NLSE. Stable stationary solitons are maintaine by the balance between the self-wavenumber ownshift, cause by the PSRS, an the upshift inuce by the linearly ecreasing SOD. The analytical solutions are foun to be in close agreement with their numerical counterparts. In this work the soliton ynamics was consiere in the moel neglecting the nonlinear ispersion an thir-orer linear ispersion. These effects will be consiere elsewhere. It may also be interesting to stuy interactions between solitons in the present moel. REFERENCES Agrawal, G.P. 1 Nonlinear Fiber Optic. Acaemic Press, San Diego. Anrianov, A., Muraviev, S., Kim, A. & Sysoliatin, A. 7 DDF-base all-fiber optical source of femtosecon pulses smoothly tune in the telecommunication range. Laser Phys

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