Generalized Tappert transformation in femtosecond nonlinear optics

Optik - International Journal for Light and Electron Optics 179 (2019) 726–732

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Original research article

Generalized Tappert transformation in femtosecond nonlinear optics

T

⁎

V.N. Serkina, , T.L. Belyaevab a b

Benemerita Universidad Autonoma de Puebla, Av. 4 Sur 104, C. P. 72001, Puebla, Mexico Universidad Autonoma del Estado de Mexico, Av. Instituto Literarion 100, C. P. 50000, Toluca, Mexico

A R T IC LE I N F O

ABS TRA CT

Keywords: The generalized higher-order nonlinear Schrödinger equation Linear external potential The generalized Tappert transformation Nonautonomous femtosecond solitons The Raman soliton self-scattering effect

We present the generalized Tappert transformation for the higher-order nonlinear Schrödinger equation model of femtosecond nonlinear optics, in which the gravitational-like potential is introduced with the aim of the effective parameterization of the Raman soliton self-scattering effect. By means of the nonlinear change of variables and nontrivial reversible gauge transformation the higher-order forced nonlinear Schrödinger equation model is turning into the “free” higher-order nonlinear Schrödinger equation without external potential, but with varying in time dispersion and nonlinearity. The proposed generalized Tappert transformations allow one to find the direct relationships between the soliton solutions of the forced nonisospectral equations and the corresponding isospectral equations without external potential, but with varying in time dispersion and nonlinearity coefficients.

1. Introduction It is well known that, for nonlinear partial differential equations (NPDEs), the linear superposition principle cannot be applied to generate new solutions. In this context, a transformation of functions and variables can sometimes be found that transforms a nonlinear equation into a linear one, or converts given NPDE to another more simple equation that is easily to resolve. It is suffice to mention two famous transformations that radically altered our understanding of nonlinear partial differential equations: the Cole–Hopf transformation [1,2] which reduces the Burgers equation to a linear diffusion equation, and the famous Miura transformation [3] which transfers every solution of the modified Korteweg–de Vries (mKdV) equation to a solution of the Korteweg–de Vries (KdV) equation. It should be mentioned in this connection an ingenious transformation that provides a remarkable possibility to convert the forced nonlinear Schrödinger equation (NLSE) with linear external potential to the canonical NLSE without potential. The forced NLSE with linear external potential V(x) = 2λ0(t)x

iq t +

1 D20 qxx + R20 |q|2 q − 2λ 0 (t )xq = 0 2

(1)

is known from the early days of the advancement of the Inverse Scattering Transform (IST) method with varying spectral parameter Λ(t) (the nonisospectral IST problem dΛ/dt = λ0(t)). As early as in 1976, Chen and Liu [4] considered the nonlinear wave propagation in inhomogeneous media in the framework of the NLSE model with external x-dependent linear potential, and they obtained exact solutions of the forced NLSE (1) in the form of accelerated solitons. In the conclusion part of their work [4], the authors have

⁎

Corresponding author. E-mail address: [email protected] (V.N. Serkin).

https://doi.org/10.1016/j.ijleo.2018.11.012 Received 7 November 2018; Accepted 7 November 2018 0030-4026/ © 2018 Elsevier GmbH. All rights reserved.

Optik - International Journal for Light and Electron Optics 179 (2019) 726–732

V.N. Serkin, T.L. Belyaeva

acknowledged Dr. Fred Tappert, who informed them that Eq. (1) can be transformed to the usual NLSE directly by means of the remarkable transformation, which is known today as the Tappert transformation. Chen and Liu in their pioneering work [4] specially emphasized that “by allowing the time-varying eigenvalue, we have therefore greatly enlarged the set of exactly solvable nonlinear time-evolution equations”. The soliton solutions in external potential governed by Eq. (1), nonautonomous solitons, interact elastically, but move with varying in time velocities (acceleration), and have variable in time spectra. In particular, the exactly integrable NLSE model with linear gravitational-like potential was applied in the sliding filter method of the noise separation from picosecond solitons both in time-space and spectral domains [5,6]. In general, the nonautonomous soliton concept [7–9] substantially extends the concept of classical solitons and generalizes it for nonautonomous nonlinear models with time-dependent external potentials and varying dispersion, nonlinearity, gain or absorption. The current state of the art in this very active and exiting field one can find, for example, in the recent publications and references therein [10–20]. In this work, we consider the generalized higher-order nonlinear Schrödinger equation (hoNLSE) model of femtosecond nonlinear optics, in which the parameterized gravitational-like potential effectively simulates the soliton Raman self-scattering effect (known also as the soliton Raman self-frequency shift). We present the gauge transformation of this models into the “free” hoNLSE without external potentials, but with varying in time dispersion and nonlinearity. Novel effects arising in this model are illustrated for the integrable version of the hoNLSE, the nonautonomous Hirota equation with linear (gravitational-like) potential, for which we demonstrate the nontrivial dynamics of its accelerating nonautonomous solitons. 2. Reversible gauge transformation of the higher-order nonlinear Schrödinger equation In femtosecond nonlinear optics, one of the basic and the most simple models is the higher-order nonlinear Schrödinger equation (hoNLSE), which includes both the influence of the higher-order linear dispersive and nonlinear dispersive effects (self-steepening), and the delayed nonlinear response – the effect of the Raman soliton self-scattering [5,6,21–26] given by the parameter σR:

iq t +

1 D20 qxx + R20 |q|2 q + iD30 qxxx + iβ2 |q|2 qx + iβ3 (|q|2 )x q − σR (|q|2 )x q = 0. 2

(2)

Eq. (2) can be transformed into the forced hoNLSE with the gravitational-like potential (with the aim of the effective parameterization of the Raman soliton self-scattering effect), if the delayed response of the nonlinearity does not affect significantly the soliton amplitude during its Raman self-scattering [9,17]:

iq t +

1 D20 qxx + R20 |q|2 q + iD30 qxxx + iβ2 |q|2 qx + iβ3 (|q|2 )x q − 2λ 0 xq = 0. 2

(3)

In Eqs. (2) and (3), the subscriptions x and t denote the derivations on x and t, and coefficients D20, R20, D30, β2, β3, are chosen, without loss of the generality, to be real constants. It should be specially emphasized that in femtosecond nonlinear-optical applications, “x → T” denotes the dimensionless time in commoving system of coordinates, and “t → Z” is the propagation distance normalized on the dispersion length. The effect of the dispersion of the nonlinearity, appearing in the framework of the Slowly Amplitude Varying method (SVA) and given by two terms with coefficients β2 and β3 in Eqs. (2) and (3). In particular, when β2 = β3 = β, the corresponding two terms in Eqs. (2) and (3) are simplified by iβ (|q|2 q)x describing just this effect [27,5]. Among the most important features of the forced hoNLSE with linear potential (3), it should be emphasized the possibility to transform this Eq. (3) into the hoNLSE without gravitational potential

iQ Z +

1 ˜ D2 (Z ) Q TT + R˜2 (Z )|Q|2 Q + iD30 Q TTT + iβ2 |Q|2 QT + iβ3 (|Q|2 )T Q = 0, 2

(4)

where the parameters of dispersion and nonlinearity acquire the following time dependences

D˜ 2 (Z ) = D20 + 12D30 λ 0 Z

and R˜2 (Z ) = R20 + 2β2 λ 0 Z .

(5)

For this purpose, we introduce a new function Q[T(x, t), Z(t)] by the gauge transformation given by

Q [T (x , t ), Z (t )] = q (x , t )exp[iφ (x , t )],

φ (x , t ) = 2λ 0 xt +

(6)

2 D20 λ 02 t 3 + 2D30 λ 03 t 4, 3

and the following change of variables

T (x , t ) = x + D20 λ 0 t 2 + 4D30 λ 02 t 3,

Z (t ) = t .

(7)

The inverse gauge transformation

q [x (T , Z ), t (T )] = Q (T , Z )exp[−iφ′ (T , Z )], φ′ (T , Z ) = 2λ 0 TZ −

(8)

4 D20 λ 02 Z 3 − 6D30 λ 03 Z 4 3

x (T , Z ) = T − D20 λ 0 Z 2 − 4D30 λ 02 Z 3,

t (Z ) = Z 727

Optik - International Journal for Light and Electron Optics 179 (2019) 726–732

V.N. Serkin, T.L. Belyaeva

transforms Eq. (4) into Eq. (3), respectively. Obviously that the transformations ((6)–(8)) can be regarded as the generalized Tappert transformations, because in the limit case of D30 = β2 = β3 = 0, the hoNLSE (3) takes the form of the forced NLSE (1). Correspondingly, the transformations ((6)–(8)) for Eq. (1) become

Q [T (x , t ), Z (t )] = q (x , t )exp[iφ (x , t )], φ (x , t ) = 2λ 0 xt +

2 D20 λ 02 t 3, 3

(9)

T (x , t ) = x + D20 λ 0 t 2,

Z (t ) = t

and

q (x , t ) = Q (T , Z )exp[−iφ′ (T , Z )], φ′ (T , Z ) = 2λ 0 TZ −

4 D20 λ 02 Z 3, 3

(10)

x = T − D20 λ 0 Z 2,

t = Z,

that is, these transformations both take the form proposed by Tappert [4] and transform the forced NLSE (1) into the canonical “free” NLSE – without potential and vise verse. The nonlinear phase and coordinates transformations ((6)–(8)) are reversible in the sense that for the “free” (without forcing ′ , R20 ′ , D30, β2, β3 potential) model with constant dispersion and nonlinearity coefficients D20

iQ Z +

1 ′ Q TT + R20 ′ |Q|2 Q + iD30 Q TTT + iβ2 |Q|2 QT + iβ3 (|Q|2 )T Q = 0, D20 2

(11)

there exists the following transformation

q (x , t ) = Q (T , Z )exp[iφ′ (T , Z )], φ′ (T , Z ) = −2λ 0 TZ +

(12)

4 ′ λ 02 Z 3 − 4D30 ′ λ 03 Z 4, D20 3

′ λ 0 Z 2 + 4D30 ′ λ 02 Z 3, x (T , Z ) = T − D20

t (Z ) = Z ,

which gives rise to the gravitational-like potential U(x) = 2λ0x and the hoNLSE

iq t +

1 D2 (t ) qxx + R2 (t )|q|2 q + iD30 qxxx + iβ2 |q|2 qx + iβ3 (|q|2 )x q − 2λ 0 xq = 0 2

(13)

with varying parameters of dispersion and nonlinearity

D2 (t ) = D2′ − 12D30 λ 0 t;

R2 (t ) = R2′ − 2β2 λ 0 t .

(14)

3. The generalized Galilean-like invariance of the higher-order nonlinear Schrödinger equation The forced generalized hoNLSE with gravitational-like potential (3) is invariant under the following nontrivial Galilean transformation

Q [T (x , t ), Z (t )] = q (x , t )exp[iS(x , t )],

S (x , t ) = −V0 x +

(15)

1 2 V0 (D20 − 2D30 V0 ) t + λ 0 V0 (D20 − 3D30 V0 ) t 2, 2

T (x , t ) = x − V0 (D20 − 3D30 V0) t ,

Z (t ) = t ,

which leaves the same form of Eq. (3) for a new function Q[T(x, t), Z(t)] dependent on the new variables T(x, t) and Z(t)

iQ Z +

1 ′ Q TT + R20 ′ |Q|2 Q + iD30 Q TTT + iβ2 |Q|2 QT + iβ3 (|Q|2 )T Q − 2λ 0 TQ = 0. D20 2

(16)

Nontriviality of this generalized Galilean transformation consists in the two following aspects. First, the phase factor S(x, t) depends now on the higher-order dispersion D30 and the acceleration constant λ0. Second, the “novel” dispersion and nonlinearity ′ and R20 ′ are connected with the “old” ones by the following algebraic relationships coefficients D20

′ = D20 − 6D30 V0, D20

′ = R20 − β2 V0, R20

(17)

where V0 is an arbitrary constant. Transformation (15)) can be considered as the generalized Galilean-like invariance and corresponds to the transition from one accelerating reference frame moving with the constant acceleration λ0 to another one moving with the same constant acceleration λ0 as well. The relative velocity between these two reference frames is given by V0(D20 − 3D30V0). If the external potential vanishes, λ0 = 0, the transformation (15)) represents a special case of the Galilean-like invariance of the higher-order “free” NLSE, which was found by Karpman [28]. Obviously, that in the limit case of the standard NLSE, when D30 = R30 = 0 and λ0 = 0, the transformation (15)) is reduced to the canonical Galilean invariance (applicable for the linear ordinary quantum-mechanical Schrödinger equation as 728

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V.N. Serkin, T.L. Belyaeva

well if R20 ≡ 0). 4. Integrable version of the high-order NLSE and nonautonomous solitons The NLSE with linear potential (1) arises in the framework of the nonisospectral IST problem [4] with varying spectral parameter Λ(t)

dΛ = λ 0 (t ), dt

⇒Λ = Λ 0 +

∫0

t

λ 0 (t ′)dt′,

(18)

where Λ(t = 0) = Λ0 = κ0 + iη0. The fundamental soliton solution of Eq. (1), the nonautonomous accelerating soliton (for simplicity, we choose here λ0(t) = λ0 = const,)

q (x , t ) = 2η0 sech[ξ (x , t )]exp[−iχ (x , t )],

(19)

ξ (x , t ) = 2η0 (x − x 0 + 2κ 0 t + λ 0 t 2), χ (x , t ) = 2(κ 0 + λ 0 t )(x − x 0) + 2(κ 02 − η02 ) t + 2κ 0 λ 0 t 2 +

2 2 3 λ0 t , 3

is characterized by the time-invariant amplitude 2η0 (given by the soliton for-factor η0) and varying in time velocity (the gravitational-like acceleration):

Λ(t ) = κ 0 + λ 0 t + iη0 .

(20)

In the gravitational-like potential U(x) = 2λ0x, the soliton moves not only in ordinary space-time, but in the spectral domain as well. The spectrum of this accelerated “colored” soliton continuously moves into the red spectral region according to the law Δω = 2λ0t. Notice that in the photonics notations, ”x → T” and ”t → Z”, the soliton red-shift is appearing to be directly proportional to the propagation distance Δω = 2λ0Z [5,6,27,29–32]. The behavior of nonautonomous solitons in the gravitational-like potential has a remarkable mathematical analogy with the soliton Raman self-scattering, which manifests itself as a continuous red-shift of the soliton frequency. Discovered in nonlinear fiber optics in Refs. [33,34], the soliton Raman self-scattering effect arises because the spectrum of a high-power ultrashort laser pulse becomes so broad that it can be continuously transformed into the red (Raman) spectral region due to the amplification of low-frequency (Stokes) components in the field of high-frequency (anti-Stokes) spectral components of the same soliton pulse. In general, the higher-order forced NLSE (3) is not integrable by the IST. One of the most important integrable versions of the higher-order NLSE follows from Eq. (3) under a special choice of the dispersion and nonlinearity coefficients β2 = 6R30 [35] and the so-called Hirota constrain: D30R20 = D20R30 [36]

iq t +

1 D20 qxx + σR20 |q|2 q + iD30 qxxx + 6iσR30 |q|2 qx − 2λ (t )xq = 0, 2

(21)

where the parameter σ = ± 1 separates bright and dark soliton solutions. Under the gauge transformation and change of variables (((6)–(8))), the forced Hirota equation (21) with linear potential can be transformed into the “free” Hirota equation with varying parameters of dispersion and nonlinearity

iQ Z +

1 ˜ D2 (Z ) Q TT + R˜2 (Z )|Q|2 Q + iD30 Q TTT + i6R30 |Q|2 QT = 0, 2

D˜ 2 (Z ) = D20 + 12D30 λ 0 Z ,

R˜2 (Z ) = R20 + 12R30 λ 0 Z ,

(22) (23)

where the Hirota condition D30 R˜2 = R30 D˜ 2 is also fulfilled. The “free” Hirota equation (22) with varying parameters of dispersion and nonlinearity (23) complies with the isospectral IST problem with a constant spectral parameter Λ(t) = Λ0 = κ0 + iη0 (this is evident by vanishing the Wronskian W (R˜2 , D˜ 2) = W (R30 , D30) = 0 [14,17]). By way of illustration, let us consider the fundamental one-soliton solution of the Hirota equation (22) with varying parameters of dispersion and nonlinearity:

Q (T , Z ) = 2η0 (D30 / R30)1/2sechξ ′ (T , Z )exp[−iχ ′ (T , Z )],

(24)

ξ ′ (T , Z ) = 2η0 [(T − T0) + 2D20 κ 0 Z + 4D30 (3κ 02 − η02 ) Z + 12D30 κ 0 λ 0 Z 2], χ ′ (T , Z ) = 2(T − T0 ) κ 0 + 2D20 (κ 02 − η02 ) Z + 8D30 κ 0 (κ 02 − 3η02 ) Z + 12D30 (κ 02 − η02 ) λ 0 Z 2. The corresponding accelerated soliton of the forced Hirota equation with linear external potential (21)) can be immediately constructed by means of the transformations ((6)–(8)):

q (x , t ) = 2η0 (D30 / R30)1/2sechξ (x , t )exp[−iχ (x , t )],

(25) 729

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V.N. Serkin, T.L. Belyaeva

Fig. 1. Dynamics of the nonautonomous two-soliton solutions of the “free” Hirota equation with varying parameters of dispersion and nonlinearity Eq. (22) after choosing the initial parameters λ0 = −0.1, D30 = R30 = 1.0, the form-factors of interacting solitons η01 = 0.5 and η02 = 0.25, and initial velocities: (a) κ01 = κ02 = 0.0; (b) κ01 = κ02 = 0.2.

ξ (x , t ) = 2η0 {(x − x 0) + 2t [D20 κ 0 + 2D30 (3κ 02 − η02)] + λ 0 (D20 + 12D30 κ 0 ) t 2 + 4D30 λ 02 t 3}, χ (x , t ) = 2(κ 0 + λ 0 t )(x − x 0) + 2t [D20 (κ 02 − η02) + 4D30 κ 0 (κ 02 − 3η02)] 2 + 2λ 0 t 2 [D20 κ 0 + 6D30 (κ 02 − η02)] + λ 02 t 3 ⎛ D20 + 8D30 κ 0⎞ + 2D30 λ 03 t 4. ⎝3 ⎠ The nonautonomous Hirota solitons ((24) and (25)) describe the propagation of ultrashort pulse in the nonuniform media [37–53], and as it follows from Eqs. (24) and (25), the higher-order dispersion D30 has critical influence on its dynamics. The most important result following directly from the solutions ((24) and (25)) consists in the fact that under the influence of the third-order dispersion, it turns out that the soliton velocity depends on its amplitude, hence, the ultra-short solitons can be separated from the noise. Mathematically, the effects of third-order dispersion and external potential are accounted for by appearing of the “acceleration” terms proportional to t2 (Z2) and t3 in the arguments ξ′(T, Z) and ξ(x, t) of the solutions ((24) and (25)). Fig. 1 shows the dynamics of the nonautonomous two-soliton solutions of the “free” Hirota equation Eq. (22) with varying parameters of dispersion and nonlinearity. At κ01 = κ02 = 0, the solitons move along straight lines with the velocities Vi = 4D30 η02i , i = 1, 2, dependent on their amplitudes and the third-order dispersion D30 (Fig. 1(a)). With nonzero initial velocities κ0i, the solitons proceed along the parabolic trajectory (Fig. 1(b)) with the velocities

Vi (Z ) = 2D20 κ 0i + 4D30 (3κ 02i − η02i ) + 24D30 κ 0i λ 0 Z ,

(26)

dependent both on their amplitudes, the dispersion coefficients D20, D30, and a parameter λ0. It should be emphasized that the “hidden” parameter of the gravitational acceleration, λ0, appeared in the dispersion and nonlinearity coefficients (23), manifests itself by the appearance of the Z-dependence of the soliton velocity in the solution given by Eq. (24). The dependence of the dynamics of the nonautonomous accelerating two-soliton solutions of the Hirota equation with linear potential Eq. (21) on the initial soliton velocities is demonstrated in Fig. 2. One can see from Eq. (25) that the soliton velocities now become

Vi (t ) = 2D20 κ 0i + 4D30 (3κ 02i − η02i ) + 2λ 0 (D20 + 12D30 κ 0i ) t + 12D30 λ 02 t 2,

(27)

and asymptotically at t ≫ 1, they do not depend on the initial values of the velocities. Thus the asymptotic behavior of the solution (25)) at t ≫ 1 is governed by the term proportional to t3 in the arguments ξ(x, t) of the solution (25)), which is clearly seen in Fig. 2. Notice that in the existing fibers, the effective relaxation time of delayed Raman nonlinearity is estimated as TR ∼ 3 ÷6 fs, so that σR = TR/T0 ≃ 0.1 corresponds to the soliton pulse durations 15 ÷ 30 fs. It is precisely this range of the pulse-width duration at that the considered higher-order effects play the most important role. 730

Optik - International Journal for Light and Electron Optics 179 (2019) 726–732

V.N. Serkin, T.L. Belyaeva

Fig. 2. Dynamics of the nonautonomous two-soliton solution of the forced Hirota equation with the linear external potential Eq. (21) after choosing the initial parameters λ0 = −0.1, D30 = R30 = 1.0, the soliton form-factors η01 = 0.5 and η02 = 0.25, and initial velocities (a) κ01 = κ02 = 0.0; (b) κ01 = κ02 = 0.2.

5. Conclusions Summarizing our findings, we have introduced the generalized Tappert transformation for the higher-order nonlinear Schrödinger equation model, in which the gravitational-like potential is introduced with the aim of the effective parameterization of the Raman soliton self-scattering effect dominating for the femtosecond optical solitons. Under the proposed transformation, the forced hoNLSE is turning into the “free” hoNLSE without external potential, but with varying in time dispersion and nonlinearity. With a certain choice of the dispersion and nonlinearity parameters, the forced hoNLSE is presented by the completely integrable physical model given by the Hirota equation with the gravitational-like potential, the novel solutions of which demonstrate that the soliton velocities are dependent both on their amplitudes and parameters of the external potentials. Thereby ultra-short solitons can be separated from the noise both in the time and spectral domains. We have illustrated arising novel effects for the femtosecond colored nonautonomous solitons in the time-range of the pulse durations 15 ÷ 30 fs. Additional open and important questions are related to the understanding to what extent our results can be applied to the stronger linear and nonlinear higher-order effects and the soliton wave-particle duality, as well as physics of soliton breathers interactions with impurities and the soliton tunneling through and trapping by external potentials [54–67]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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