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Determinant solutions and asymptotic state analysis for an integrable model of transient stimulated Raman scattering
Optik - International Journal for Light and Electron Optics 200 (2020) 163348
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Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Determinant solutions and asymptotic state analysis for an integrable model of transient stimulated Raman scattering Xiang-Hua Menga, Xiao-Yong Wena, Linhua Piaoa,b, Deng-Shan Wanga, a b
T
⁎
School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China Research Center for Sensor Technology, Beijing Information Science and Technology University, Jianxiangqiao Campus, Beijing 100101, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Transient stimulated Raman scattering equations Darboux transformation Explicit determinant soliton solution Asymptotic state analysis
In this paper, the transient stimulated Raman scattering equations, which describe the stimulated Raman scattering in the transient limit, are investigated by Darboux transformation. Considering the trivial initial solutions, the explicit N-soliton solutions in determinant form for the transient stimulated Raman scattering equations are presented. Based on the explicit solutions formula, the one- and two-soliton solutions are given explicitly. The asymptotic state analysis of the twosoliton solutions is conducted. And different soliton propagation phenomena are discussed and illustrated including but not limited to the flat-top soliton, the two-hump soliton and bound two soliton with periodic structure, which is good for understanding the interaction properties under different wave parameters.
1. Introduction The nonlinear evolution equations have been attracting enormous attention since they can describe significant nonlinear phenomena in different fields including the optical fiber communication [1,2], Bose-Einstein condensation [3] and fluids dynamics [4–6]. Using the soliton theory, a wide range of nonlinear evolution equations have been found exhibiting solitary wave solutions [7–9], and abundant solitary waves structures have been found including line solitons, breather, lump and rogue waves [10–12]. Stimulated Raman scattering (SRS), first observed by Woodybury and Ng [13], has important applications to spectroscopy, beam cleanup and combination [14]. Neglecting diffraction, level saturation and higher-order Stokes generation from spontaneous emission, the following coupled equations for the stimulated Raman scattering in the transient limit was presented [15]
qt = A1 A2* ,
A1x = −A2 q,
A2x = A1 q*,
(1)
with A1, A2 and q all complex functions of x and t which corresponds to the pump electric field, the Stokes electric field and the material excitation wave. Under the transformations defined by
p = 2 i A1 A2* ,
w = |A1 |2 − |A2 |2 ,
(2)
the equations for q, p and w are given as
qt = −
i p, 2
px = 2 i q w,
wx = i (p q* − q p*),
(3)
where both q = q(x, t) and p = p(x, t) are complex functions, and w = w (x , t ) is a real one. The transient stimulated Raman scattering
⁎
Corresponding author. E-mail address: [email protected] (D.-S. Wang).
https://doi.org/10.1016/j.ijleo.2019.163348 Received 30 August 2019; Accepted 31 August 2019 0030-4026/ © 2019 Published by Elsevier GmbH.
Optik - International Journal for Light and Electron Optics 200 (2020) 163348
X.-H. Meng, et al.
(TSRS) equations (3) are integrable in the sense of admitting the Lax representation. Based on the previous literatures, solitons in the SRS have been investigated by virtue of theoretical and experimental studies [16,17]. The initial-boundary value problems for Eq. (3) have been discussed [15,18]. Moreover, several concerns have been on the SRS equations with damping terms have been studied [14,17]. However, the explicit N-soliton solutions for the TSRS equations (3) have not been constructed up to now. Among the analytical methods for solving the nonlinear evolution equations [19–25], Darboux transformation and its improved form have been effective tools for seeking the explicit multi-soliton solutions based on the Lax pair of equations [26–28]. In this paper, by means of the Darboux transformation, the explicit N-soliton solutions is presented in determinant form. In addition, the asymptotic state analysis [29–32] of two-soliton solutions is conducted so that the elastic interaction properties between two solitons of the TSRS equation are given. Different soliton propagation phenomena are demonstrated and analyzed. 2. Darboux transformation and determinant soliton solutions The Lax pair [15] for the TSRS equations is
Ψx = U Ψ,
Ψ = (ψ1, ψ2 )T ,
(4)
1 ⎛i w − p ⎞ , 4 λ ⎝ p* − i w ⎠
(5)
Ψt = V Ψ,
−iλ q⎞ U = ⎜⎛ ⎟, ⎝ − q* i λ ⎠
V=
⎜
⎟
where λ is a complex constant referred to the spectral parameter. Constructing the gauge transformation,
ˆ = T Ψ, Ψ
(6)
ˆ satisfies the Lax pair of the same form with Ψ, i.e. where Ψ
ˆx = U ˆΨ ˆ, Ψ
ˆt = V ˆΨ ˆ, Ψ
(7)
ˆ = (Tx + T U ˆ ) T −1, U
ˆ = (Tx + T V ˆ ) T −1. V
(8)
Expanding T as a first order polynomial of λ, T = λ A + B with A and B both as undetermined 2 × 2 matrices. By virtue of the condition T Ψ[0] |λ = λ1 = 0 , the one-fold Darboux transformation for the TSRS equations can be constructed as
Ψ[1] = T[1] Ψ[0], T[1] = λ
q[1] = q[0] − 2 i b0 ,
(10 01) + ⎛⎝ac
c0 =
1 D2
1 D2
1 g[0]
1* λ1*g[0]
−
(9)
⎟
1* λ1 f[0]
1 f[0]
w[1] = w[0] − 4 i a0t ,
b0 ⎞ , d0 ⎠
0
⎜
0
a0 =
p[1] = p[0] + 4 b0t ,
1 1 g[0] λ1 g[0] 1* 1* f[0] λ1*f[0]
,
,
(10)
b0 =
d0 =
1 1 f[0] λ1 f[0] 1 , 1 1* D2 − g[0]* − λ1*g[0]
1 1 f[0] λ1 g[0] 1 , 1* 1* D2 − g[0] λ1*f[0]
D2 =
(11) 1 f[0]
1 g[0]
.
1* 1* − g[0] f[0]
With the initial solutions of the TSRS equations q[0], p[0] and
(12)
1 w[0], (f[0] ,
1 T g[0] )
=
1 Ψ[0]
is a special solution of the Lax pair (4)
1* 1* T , f[0] ) is the one corresponding to the spectral parameter λ1* . Iterating N-times, corresponding to the spectral parameter λ1 and (−g[0] the N-fold Darboux transformation can be obtained, based on which the N-soliton solution can be given with trivial initial solutions as
q[N ] = q[0] − 2 i
A2N , D2N
(13)
A p[N ] = p[0] + 4 ⎛ 2N ⎞ , ⎝ D2N ⎠t
(14)
B w[N ] = w[0] − 4 i ⎛ 2N ⎞ , D ⎝ 2N ⎠t
(15)
⎜
⎟
⎜
⎟
where 1 f[0]
A2N =
1* − g[0] ⋯ N f[0]
1 g[0]
1 λ1 f[0]
1 λ1 g[0] ⋯
1 λ1N − 1 f[0]
1 λ1N f[0]
1* 1* 1* 1* 1* f[0] − λ1* g[0] λ1* f[0] ⋯ − λ1*N − 1 g[0] − λ1*N g[0] ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ N N N N N g[0] λN f[0] λN g[0] ⋯ λNN − 1 f[0] λNN f[0]
N* N* N* N* N* N* − g[0] f[0] − λN* g[0] λN* f[0] ⋯ − λN*N − 1 g[0] − λN*N g[0]
2
,
(16)
Optik - International Journal for Light and Electron Optics 200 (2020) 163348
X.-H. Meng, et al. 1 f[0] 1* g[0]
− ⋯ N f[0]
B2N =
1 g[0]
1 λ1 f[0]
1 1 λ1 g[0] ⋯ λ1N − 2 g[0]
1 λ1N − 1 g[0]
1 λ1N f[0]
1* f[0]
1* λ1* g[0]
1* λ1* f[0]
1* λ1*N − 1 f[0]
1* − λ1*N g[0] ⋯ N λNN f[0]
−
⋯ N g[0]
⋯ N λN f[0]
⋯ N λN g[0]
1* λ1*N − 2 f[0]
⋯ ⋯ ⋯ N ⋯ λNN − 2 g[0]
⋯ N λNN − 1 f[0]
,
N* N* N* N* N* N* N* − g[0] f[0] − λN* g[0] λN* f[0] ⋯ λN*N − 2 f[0] λN*N − 1 f[0] − λN*N g[0] 1 f[0]
1 g[0]
1* − g[0] ⋯ N f[0]
D2N =
1 λ1 f[0]
1 λ1 g[0] ⋯
1 λ1N − 1 f[0]
(17)
1 λ1N − 1 g[0]
1* 1* 1* 1* 1* f[0] − λ1* g[0] λ1* f[0] ⋯ − λ1*N − 1 g[0] λ1*N − 1 f[0] ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ N N N N N g[0] λN f[0] λN g[0] ⋯ λNN − 1 f[0] λNN − 1 g[0]
,
N* N* N* N* N* N* − g[0] f[0] − λN* g[0] λN* f[0] ⋯ − λN*N − 1 g[0] λN*N − 1 f[0]
(18)
= is a special solution of the Lax pair corresponding to the spectral parameter λN and the initial seed solution of N* N* T the TSRS equation, and (−g[0] , f[0] ) is a special solution of the Lax pair corresponding to the spectral parameter λN* and seed solution.
where
N (f[0] ,
N T g[0] )
N Ψ[0]
3. Soliton solutions and asymptotic state analysis For seeking soliton solutions of the TSRS equations (3), the following initial solutions are taken
q[0] = 0,
p[0] = 0,
w[0] = ω0 ,
(19)
with ω0 as a nonzero real constant. The substitution of the initial solutions into the Lax pair can yield
ψ1[0] = e−i λ x +
i ω0 t 4λ ,
ψ2[0] = ei λ x −
i ω0 t 4λ .
(20)
For the special solutions with spectral parameter λ1, 1 f[0] =e
−i λ1 x +
i ω0 t 4 λ1 ,
1 g[0] =e
i λ1 x −
i ω0 t 4 λ1 .
(21)
By virtue of the one-fold Darboux transformation, the one-soliton solutions for the TSRS equations are given as below
q[1] = 2 λ1I sech θ1 e−i ζ1, p[1] =
(22)
−2 ω0 λ1I sechθ1 (λ1R + i λ1I tanhθ1) e−i ζ1, |λ1 |2
w[1] = ω0 −
(23)
2 ω0 λ12I sech2 θ1, |λ1 |2
(24)
with
ω0 ⎞ θ1 = 2λ1I ⎛x + t , 4 |λ1 |2 ⎠ ⎝ ⎜
⎟
ω0 ⎞ ζ1 = 2λ1R ⎛x − t . 4 |λ1 |2 ⎠ ⎝ ⎜
⎟
(25)
Based on the solution (13)–(15) and introducing the notations below,
ω0 ⎞ θ1 = 2λ1I ⎛x + t , 4 |λ1 |2 ⎠ ⎝
ω0 ⎞ ζ1 = 2λ1R ⎛x − t , 4 |λ1 |2 ⎠ ⎝
(26)
ω0 ⎞ θ2 = 2λ2I ⎛x + t , 4 |λ2 |2 ⎠ ⎝
ω0 ⎞ ζ2 = 2λ2R ⎛x − t , 4 |λ2 |2 ⎠ ⎝
(27)
⎜
⎜
⎟
⎟
⎜
⎜
⎟
⎟
the two-soliton solutions for the TSRS equations can be given explicitly as
(2 Λ2 coshθ2 − 2 i Λ1 sinhθ2) e−i ζ1 + (2 Λ3 coshθ1 e−i ζ2 + 2 i Λ1 sinhθ1) e−i ζ2 , H1 cosh(θ1 + θ2) + H2 cosh(θ1 − θ2) − 4 λ1I λ2I cos(ζ1 − ζ2)
(28)
(4 i Λ2 coshθ2 + 4 Λ1 sinhθ2) e−i ζ1 + (4 i Λ3 coshθ1 e−i ζ2 − 4 Λ1 sinhθ1) e−i ζ2 ⎤ p=⎡ , ⎢ ⎥ H1 cosh(θ1 + θ2) + H2 cosh(θ1 − θ2) − 4 λ1I λ2I cos(ζ1 − ζ2) ⎣ ⎦t
(29)
wnum w = ω0 + , [H1 cosh(θ1 + θ2) + H2 cosh(θ1 − θ2) − 4 λ1I λ2I cos(ζ1 − ζ2)]2
(30)
q=
3
Optik - International Journal for Light and Electron Optics 200 (2020) 163348
X.-H. Meng, et al.
wnum = 2 h1 cosh(2θ1) + 2 h2 cosh(2θ2) + 4 [h3R sinh(θ1 + θ2) + h4R sinh(θ1 − θ2)] × sin(ζ1 − ζ2) + 4 [h3I cosh(θ1 + θ2) + h4I cosh(θ1 − θ2)] cos(ζ1 − ζ2),
(31)
where
H1 = (λ1R − λ2R )2 + (λ1I − λ2I )2 , Λ1 = 4(λ1R − λ2R ) λ1I λ2I ,
H2 = (λ1R − λ2R )2 + (λ1I + λ2I )2 ,
Λ2 = 2λ1I [(λ1R − λ2R )2 + λ12I − λ 22I ],
Λ3 = 2λ2I [(λ1R − λ2R )2 + λ12I − λ 22I ], and
−2λ 22I [(λ1R − λ2R )2 + λ12I + λ 22I ]2 ω0 + 8λ12I λ 24I ω0 , |λ2 |2 −2λ12I [(λ1R − λ2R )2 + λ12I + λ 22I ]2 ω0 + 8λ14I λ 22I ω0 , h2 = |λ1 |2 2 2iH1 λ1I λ2I (−iλ1R + λ1I + i λ2R + λ2I ) ω0 h3 = = h3R + ih3I , (λ1R + iλ1I )(λ2R − iλ2I ) 2iH2 λ1I λ2I (iλ1R − λ1I − iλ2R + λ2I )2ω0 h4 = = h4R + ih4I . (λ1R + iλ1I )(λ2R + iλ2I ) h1 =
In the following, the asymptotic state analysis of the two-soliton solution will be given for t→ ± ∞. Without loss of generality, let us assume w0 (|λ1 |2 − |λ2 |2 ) > 0, λ1I > 0 and λ2I > 0. The asymptotic states of solitons (28)–(30) are
r1 ⎧ sech(θ1 + δ1) e−i (ζ1− ϕ1), θ1 ∼ O (1), t → − ∞ , ⎪ H1 H2 q→ ⎨ r1 sech(θ1 − δ1) e−i (ζ1+ ϕ1), θ1 ∼ O (1), t → + ∞ , ⎪ H1 H2 ⎩
(32)
r ω ˆ ⎧ 21 0 sech(θ1 + δ1)[λ1R − iλ1I tanh(θ1 + δ1)] e−i (ζ1− ϕ1), θ1 ∼ O (1), t → − ∞ , ⎪ |λ1 | H1 H2 p→ r1 ω0 ˜ ⎨ sech(θ1 − δ1)[λ1R − iλ1I tanh(θ1 − δ1)] e−i (ζ1− ϕ1), θ1 ∼ O (1), t → + ∞ , ⎪ |λ1 |2 H1 H2 ⎩
(33)
⎧ w0 + 4 h1 sech2 (θ1 + δ1), θ1 ∼ O (1), t → − ∞ , ⎪ H1 H2 w→ 4 h1 ⎨ + w sech2 (θ1 − δ1), θ1 ∼ O (1), t → + ∞ , ⎪ 0 H 1 H2 ⎩
(34)
where
r1 =
4λ12I [λ12I + (λ1R − λ2R )2 + λ 22I ]2 − 16λ14I λ 22I ,
ϕ1 = arg(Λ2 + i Λ1),
δ1 = ln
ϕˆ1 = arg(−Λ2 − i Λ1) = ± π + ϕ1,
H2 , H1 ϕ˜1 = arg(−Λ2 + i Λ1) = ± π − ϕ1.
(35) (36)
From the asymptotic state analysis of two-soliton solution, it can be seen that the elastic interaction property between the two solitons is preserved. Following the same procedure, the asymptotic state solutions for θ2 ∼ O(1) when t→ ± ∞ are
r2 ⎧ sech(θ2 − δ1) e−i (ζ2− ϕ2), t → − ∞ , ⎪ H1 H2 q→ ⎨ r2 sech(θ2 + δ1) e−i (ζ2+ ϕ2), t → + ∞ , ⎪ H1 H2 ⎩
(37)
r ω ˆ ⎧ 22 0 sech(θ2 − δ1)[λ2R + iλ2I tanh(θ2 − δ1)] e−i (ζ1− ϕ2), t → − ∞ , ⎪ |λ2 | H1 H2 p→ r2 ω0 ˜ ⎨ sech(θ2 + δ1)[λ2R − iλ2I tanh(θ2 + δ1)] e−i (ζ2− ϕ2), t → + ∞ , ⎪ |λ2 |2 H1 H2 ⎩
(38)
⎧ w0 + 4 h2 sech2 (θ2 − δ1), t → − ∞ , ⎪ H1 H2 w→ 4 h2 ⎨ 2 ⎪ w0 + H H sech (θ2 + δ1), t → + ∞ , 1 2 ⎩
(39)
where 4
Optik - International Journal for Light and Electron Optics 200 (2020) 163348
X.-H. Meng, et al.
Fig. 1. Plots of the one-soliton solution in (a1)–(c1) with parameters: ω0 = −0.5, λ1 = 0.3 + 0.3 i.
r2 = 4λ42 [λ 22 + (λ1 − λ3 )2 + λ42 ]2 − 16λ 22 λ44 ,
ϕ2 = arg(Λ3 + i Λ1),
ϕˆ2 = arg(−Λ3 − i Λ1) = ± π + ϕ2 ,
(40)
ϕ˜2 = arg(−Λ3 + i Λ1) = ± π − ϕ2 .
(41)
4. Dynamical properties of the soliton solutions In this paper, the multi-soliton solutions in determinant form for the TSRS equations have been obtained using the Darboux transformation. In addition, the asymptotic state solutions for the two solitons have been analyzed, which indicates the interaction between the two solitons is elastic when |λ1|2 ≠ |λ2|2. In the following, by choosing different spectral parameters and ω0, abundant soliton evolution situations will be discussed. The positive and negative parameter ω0 corresponds to the anti-bell and bell shape soliton of w component, and the absolute value of ω0 affects the soliton width of the three components. The bigger the absolute value of ω0 is, the smaller the soliton width is. For different spectral parameter condition, two different soliton structures of p component will appear. As seen from Figs. 1(b1) and 2(b2) , when the absolute value of the real part of the spectral parameter is the same with the absolute value of the imaginary part, i.e. |λ1R| = |λ1I|, the soliton |p| is with a flat top. A two-hump soliton |p| will be shown as Figs. 3(b3) and 4(b4) when |λ1R| < |λ1I|. While a normal bell shape soliton |p| appears when the absolute value of the real part of the spectral parameter is larger than the absolute value of its imaginary part, i.e. |λ1R| > |λ1I|. Abundant two-soliton structures can be found for different spectral parameters. For two pure imaginary spectral parameters, seen from Fig. 5(a5)–(c5) , the two solitons interact with each other, and separate remaining their own original shape, while in Fig. 6(a6)–(c6) , two parallel soliton propagation phenomena can be seen. For two pure imaginary spectral parameters λ1 and λ2, the absolute value of the real part of the spectral parameter is less than the absolute value of its imaginary part, the two solitons of |p| are both two-hump solitons. When λ1 = −λ 2* with nonzero real and imaginary parts, the bound two soliton interaction occurs seen from Fig. 7(a7)–(c7) , and solitons propagation of the three components present periodical feature. Fig. 8(a8)–(c8) demonstrates the two soliton interaction with one pure imaginary spectral parameter and one complex spectral parameter with the same real and imaginary parts. From the figures, it can be seen that the |p| soliton consists of one flat-top soliton and one two-hump soliton. 5. Conclusions Based on the Darboux transformation, the multi-soliton solutions for the TSRS equations have been obtained in determinant forms. For different spectral parameters, some interesting soliton interaction phenomena have been demonstrated. More soliton structures can be illustrated for corresponding spectral parameters and the occurrence condition of different soliton structures can be analysed from the analytic solutions. The results obtained in this paper are hoped to be applicable in a variety of physical problems including the simulated Raman scattering.
Fig. 2. The corresponding section plots of the one-soliton solution in Fig. 1 at t = 0. 5
Optik - International Journal for Light and Electron Optics 200 (2020) 163348
X.-H. Meng, et al.
Fig. 3. Plots of the one-soliton solution in (a3)–(c3) with parameters: ω0 = 1, λ1 = 0.2 + 0.5 i.
Fig. 4. The corresponding section plots of the one-soliton solution in Fig. 3 at t = 0.
Fig. 5. Plots of the two-soliton solution with parameters: ω0 = 1, λ1 = 0.3 i, λ2 = 0.5 i.
Conflict of interest There is no potential conflict of interest.
Acknowledgements This work has been supported by the National Natural Science Foundation of China under Grant Nos. 11401031 and 11971067, Beijing Natural Science Foundation under Grant No. 1182009, the Beijing Great Wall Talents Cultivation Program under Grant No. CIT&TCD20180325, Qin Xin Talents Cultivation Program of Beijing Information Science and Technology University (QXTCPB201704 and QXTCP A201702Scientific Research Common Program of Beijing Municipal Commission of Education under Grant No. KM201911232011 and Modern measurement and control technology ministry of education key laboratory funding, the project of promoting the development of University connotation of Beijing Information Science and Technology University (5121911032, 5211910965, 5121911031 and 5221835201). 6
Optik - International Journal for Light and Electron Optics 200 (2020) 163348
X.-H. Meng, et al.
Fig. 6. Plots of the two-soliton solution with parameters: ω0 = 1, λ1 = −0.4999 i, λ2 = 0.5 i.
Fig. 7. Plots of the two-soliton solution with parameters: ω0 = 1, λ1 = 0.3 + 0.4 i, λ2 = −0.3 + 0.4 i.
Fig. 8. Plots of the two-soliton solution with parameters: ω0 = 1, λ1 = 0.5 + 0.5 i, λ2 = 0.4 i.
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Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Determinant solutions and asymptotic state analysis for an integrable model of transient stimulated Raman scattering Xiang-Hua Menga, Xiao-Yong Wena, Linhua Piaoa,b, Deng-Shan Wanga, a b
T
⁎
School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China Research Center for Sensor Technology, Beijing Information Science and Technology University, Jianxiangqiao Campus, Beijing 100101, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Transient stimulated Raman scattering equations Darboux transformation Explicit determinant soliton solution Asymptotic state analysis
In this paper, the transient stimulated Raman scattering equations, which describe the stimulated Raman scattering in the transient limit, are investigated by Darboux transformation. Considering the trivial initial solutions, the explicit N-soliton solutions in determinant form for the transient stimulated Raman scattering equations are presented. Based on the explicit solutions formula, the one- and two-soliton solutions are given explicitly. The asymptotic state analysis of the twosoliton solutions is conducted. And different soliton propagation phenomena are discussed and illustrated including but not limited to the flat-top soliton, the two-hump soliton and bound two soliton with periodic structure, which is good for understanding the interaction properties under different wave parameters.
1. Introduction The nonlinear evolution equations have been attracting enormous attention since they can describe significant nonlinear phenomena in different fields including the optical fiber communication [1,2], Bose-Einstein condensation [3] and fluids dynamics [4–6]. Using the soliton theory, a wide range of nonlinear evolution equations have been found exhibiting solitary wave solutions [7–9], and abundant solitary waves structures have been found including line solitons, breather, lump and rogue waves [10–12]. Stimulated Raman scattering (SRS), first observed by Woodybury and Ng [13], has important applications to spectroscopy, beam cleanup and combination [14]. Neglecting diffraction, level saturation and higher-order Stokes generation from spontaneous emission, the following coupled equations for the stimulated Raman scattering in the transient limit was presented [15]
qt = A1 A2* ,
A1x = −A2 q,
A2x = A1 q*,
(1)
with A1, A2 and q all complex functions of x and t which corresponds to the pump electric field, the Stokes electric field and the material excitation wave. Under the transformations defined by
p = 2 i A1 A2* ,
w = |A1 |2 − |A2 |2 ,
(2)
the equations for q, p and w are given as
qt = −
i p, 2
px = 2 i q w,
wx = i (p q* − q p*),
(3)
where both q = q(x, t) and p = p(x, t) are complex functions, and w = w (x , t ) is a real one. The transient stimulated Raman scattering
⁎
Corresponding author. E-mail address: [email protected] (D.-S. Wang).
https://doi.org/10.1016/j.ijleo.2019.163348 Received 30 August 2019; Accepted 31 August 2019 0030-4026/ © 2019 Published by Elsevier GmbH.
Optik - International Journal for Light and Electron Optics 200 (2020) 163348
X.-H. Meng, et al.
(TSRS) equations (3) are integrable in the sense of admitting the Lax representation. Based on the previous literatures, solitons in the SRS have been investigated by virtue of theoretical and experimental studies [16,17]. The initial-boundary value problems for Eq. (3) have been discussed [15,18]. Moreover, several concerns have been on the SRS equations with damping terms have been studied [14,17]. However, the explicit N-soliton solutions for the TSRS equations (3) have not been constructed up to now. Among the analytical methods for solving the nonlinear evolution equations [19–25], Darboux transformation and its improved form have been effective tools for seeking the explicit multi-soliton solutions based on the Lax pair of equations [26–28]. In this paper, by means of the Darboux transformation, the explicit N-soliton solutions is presented in determinant form. In addition, the asymptotic state analysis [29–32] of two-soliton solutions is conducted so that the elastic interaction properties between two solitons of the TSRS equation are given. Different soliton propagation phenomena are demonstrated and analyzed. 2. Darboux transformation and determinant soliton solutions The Lax pair [15] for the TSRS equations is
Ψx = U Ψ,
Ψ = (ψ1, ψ2 )T ,
(4)
1 ⎛i w − p ⎞ , 4 λ ⎝ p* − i w ⎠
(5)
Ψt = V Ψ,
−iλ q⎞ U = ⎜⎛ ⎟, ⎝ − q* i λ ⎠
V=
⎜
⎟
where λ is a complex constant referred to the spectral parameter. Constructing the gauge transformation,
ˆ = T Ψ, Ψ
(6)
ˆ satisfies the Lax pair of the same form with Ψ, i.e. where Ψ
ˆx = U ˆΨ ˆ, Ψ
ˆt = V ˆΨ ˆ, Ψ
(7)
ˆ = (Tx + T U ˆ ) T −1, U
ˆ = (Tx + T V ˆ ) T −1. V
(8)
Expanding T as a first order polynomial of λ, T = λ A + B with A and B both as undetermined 2 × 2 matrices. By virtue of the condition T Ψ[0] |λ = λ1 = 0 , the one-fold Darboux transformation for the TSRS equations can be constructed as
Ψ[1] = T[1] Ψ[0], T[1] = λ
q[1] = q[0] − 2 i b0 ,
(10 01) + ⎛⎝ac
c0 =
1 D2
1 D2
1 g[0]
1* λ1*g[0]
−
(9)
⎟
1* λ1 f[0]
1 f[0]
w[1] = w[0] − 4 i a0t ,
b0 ⎞ , d0 ⎠
0
⎜
0
a0 =
p[1] = p[0] + 4 b0t ,
1 1 g[0] λ1 g[0] 1* 1* f[0] λ1*f[0]
,
,
(10)
b0 =
d0 =
1 1 f[0] λ1 f[0] 1 , 1 1* D2 − g[0]* − λ1*g[0]
1 1 f[0] λ1 g[0] 1 , 1* 1* D2 − g[0] λ1*f[0]
D2 =
(11) 1 f[0]
1 g[0]
.
1* 1* − g[0] f[0]
With the initial solutions of the TSRS equations q[0], p[0] and
(12)
1 w[0], (f[0] ,
1 T g[0] )
=
1 Ψ[0]
is a special solution of the Lax pair (4)
1* 1* T , f[0] ) is the one corresponding to the spectral parameter λ1* . Iterating N-times, corresponding to the spectral parameter λ1 and (−g[0] the N-fold Darboux transformation can be obtained, based on which the N-soliton solution can be given with trivial initial solutions as
q[N ] = q[0] − 2 i
A2N , D2N
(13)
A p[N ] = p[0] + 4 ⎛ 2N ⎞ , ⎝ D2N ⎠t
(14)
B w[N ] = w[0] − 4 i ⎛ 2N ⎞ , D ⎝ 2N ⎠t
(15)
⎜
⎟
⎜
⎟
where 1 f[0]
A2N =
1* − g[0] ⋯ N f[0]
1 g[0]
1 λ1 f[0]
1 λ1 g[0] ⋯
1 λ1N − 1 f[0]
1 λ1N f[0]
1* 1* 1* 1* 1* f[0] − λ1* g[0] λ1* f[0] ⋯ − λ1*N − 1 g[0] − λ1*N g[0] ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ N N N N N g[0] λN f[0] λN g[0] ⋯ λNN − 1 f[0] λNN f[0]
N* N* N* N* N* N* − g[0] f[0] − λN* g[0] λN* f[0] ⋯ − λN*N − 1 g[0] − λN*N g[0]
2
,
(16)
Optik - International Journal for Light and Electron Optics 200 (2020) 163348
X.-H. Meng, et al. 1 f[0] 1* g[0]
− ⋯ N f[0]
B2N =
1 g[0]
1 λ1 f[0]
1 1 λ1 g[0] ⋯ λ1N − 2 g[0]
1 λ1N − 1 g[0]
1 λ1N f[0]
1* f[0]
1* λ1* g[0]
1* λ1* f[0]
1* λ1*N − 1 f[0]
1* − λ1*N g[0] ⋯ N λNN f[0]
−
⋯ N g[0]
⋯ N λN f[0]
⋯ N λN g[0]
1* λ1*N − 2 f[0]
⋯ ⋯ ⋯ N ⋯ λNN − 2 g[0]
⋯ N λNN − 1 f[0]
,
N* N* N* N* N* N* N* − g[0] f[0] − λN* g[0] λN* f[0] ⋯ λN*N − 2 f[0] λN*N − 1 f[0] − λN*N g[0] 1 f[0]
1 g[0]
1* − g[0] ⋯ N f[0]
D2N =
1 λ1 f[0]
1 λ1 g[0] ⋯
1 λ1N − 1 f[0]
(17)
1 λ1N − 1 g[0]
1* 1* 1* 1* 1* f[0] − λ1* g[0] λ1* f[0] ⋯ − λ1*N − 1 g[0] λ1*N − 1 f[0] ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ N N N N N g[0] λN f[0] λN g[0] ⋯ λNN − 1 f[0] λNN − 1 g[0]
,
N* N* N* N* N* N* − g[0] f[0] − λN* g[0] λN* f[0] ⋯ − λN*N − 1 g[0] λN*N − 1 f[0]
(18)
= is a special solution of the Lax pair corresponding to the spectral parameter λN and the initial seed solution of N* N* T the TSRS equation, and (−g[0] , f[0] ) is a special solution of the Lax pair corresponding to the spectral parameter λN* and seed solution.
where
N (f[0] ,
N T g[0] )
N Ψ[0]
3. Soliton solutions and asymptotic state analysis For seeking soliton solutions of the TSRS equations (3), the following initial solutions are taken
q[0] = 0,
p[0] = 0,
w[0] = ω0 ,
(19)
with ω0 as a nonzero real constant. The substitution of the initial solutions into the Lax pair can yield
ψ1[0] = e−i λ x +
i ω0 t 4λ ,
ψ2[0] = ei λ x −
i ω0 t 4λ .
(20)
For the special solutions with spectral parameter λ1, 1 f[0] =e
−i λ1 x +
i ω0 t 4 λ1 ,
1 g[0] =e
i λ1 x −
i ω0 t 4 λ1 .
(21)
By virtue of the one-fold Darboux transformation, the one-soliton solutions for the TSRS equations are given as below
q[1] = 2 λ1I sech θ1 e−i ζ1, p[1] =
(22)
−2 ω0 λ1I sechθ1 (λ1R + i λ1I tanhθ1) e−i ζ1, |λ1 |2
w[1] = ω0 −
(23)
2 ω0 λ12I sech2 θ1, |λ1 |2
(24)
with
ω0 ⎞ θ1 = 2λ1I ⎛x + t , 4 |λ1 |2 ⎠ ⎝ ⎜
⎟
ω0 ⎞ ζ1 = 2λ1R ⎛x − t . 4 |λ1 |2 ⎠ ⎝ ⎜
⎟
(25)
Based on the solution (13)–(15) and introducing the notations below,
ω0 ⎞ θ1 = 2λ1I ⎛x + t , 4 |λ1 |2 ⎠ ⎝
ω0 ⎞ ζ1 = 2λ1R ⎛x − t , 4 |λ1 |2 ⎠ ⎝
(26)
ω0 ⎞ θ2 = 2λ2I ⎛x + t , 4 |λ2 |2 ⎠ ⎝
ω0 ⎞ ζ2 = 2λ2R ⎛x − t , 4 |λ2 |2 ⎠ ⎝
(27)
⎜
⎜
⎟
⎟
⎜
⎜
⎟
⎟
the two-soliton solutions for the TSRS equations can be given explicitly as
(2 Λ2 coshθ2 − 2 i Λ1 sinhθ2) e−i ζ1 + (2 Λ3 coshθ1 e−i ζ2 + 2 i Λ1 sinhθ1) e−i ζ2 , H1 cosh(θ1 + θ2) + H2 cosh(θ1 − θ2) − 4 λ1I λ2I cos(ζ1 − ζ2)
(28)
(4 i Λ2 coshθ2 + 4 Λ1 sinhθ2) e−i ζ1 + (4 i Λ3 coshθ1 e−i ζ2 − 4 Λ1 sinhθ1) e−i ζ2 ⎤ p=⎡ , ⎢ ⎥ H1 cosh(θ1 + θ2) + H2 cosh(θ1 − θ2) − 4 λ1I λ2I cos(ζ1 − ζ2) ⎣ ⎦t
(29)
wnum w = ω0 + , [H1 cosh(θ1 + θ2) + H2 cosh(θ1 − θ2) − 4 λ1I λ2I cos(ζ1 − ζ2)]2
(30)
q=
3
Optik - International Journal for Light and Electron Optics 200 (2020) 163348
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wnum = 2 h1 cosh(2θ1) + 2 h2 cosh(2θ2) + 4 [h3R sinh(θ1 + θ2) + h4R sinh(θ1 − θ2)] × sin(ζ1 − ζ2) + 4 [h3I cosh(θ1 + θ2) + h4I cosh(θ1 − θ2)] cos(ζ1 − ζ2),
(31)
where
H1 = (λ1R − λ2R )2 + (λ1I − λ2I )2 , Λ1 = 4(λ1R − λ2R ) λ1I λ2I ,
H2 = (λ1R − λ2R )2 + (λ1I + λ2I )2 ,
Λ2 = 2λ1I [(λ1R − λ2R )2 + λ12I − λ 22I ],
Λ3 = 2λ2I [(λ1R − λ2R )2 + λ12I − λ 22I ], and
−2λ 22I [(λ1R − λ2R )2 + λ12I + λ 22I ]2 ω0 + 8λ12I λ 24I ω0 , |λ2 |2 −2λ12I [(λ1R − λ2R )2 + λ12I + λ 22I ]2 ω0 + 8λ14I λ 22I ω0 , h2 = |λ1 |2 2 2iH1 λ1I λ2I (−iλ1R + λ1I + i λ2R + λ2I ) ω0 h3 = = h3R + ih3I , (λ1R + iλ1I )(λ2R − iλ2I ) 2iH2 λ1I λ2I (iλ1R − λ1I − iλ2R + λ2I )2ω0 h4 = = h4R + ih4I . (λ1R + iλ1I )(λ2R + iλ2I ) h1 =
In the following, the asymptotic state analysis of the two-soliton solution will be given for t→ ± ∞. Without loss of generality, let us assume w0 (|λ1 |2 − |λ2 |2 ) > 0, λ1I > 0 and λ2I > 0. The asymptotic states of solitons (28)–(30) are
r1 ⎧ sech(θ1 + δ1) e−i (ζ1− ϕ1), θ1 ∼ O (1), t → − ∞ , ⎪ H1 H2 q→ ⎨ r1 sech(θ1 − δ1) e−i (ζ1+ ϕ1), θ1 ∼ O (1), t → + ∞ , ⎪ H1 H2 ⎩
(32)
r ω ˆ ⎧ 21 0 sech(θ1 + δ1)[λ1R − iλ1I tanh(θ1 + δ1)] e−i (ζ1− ϕ1), θ1 ∼ O (1), t → − ∞ , ⎪ |λ1 | H1 H2 p→ r1 ω0 ˜ ⎨ sech(θ1 − δ1)[λ1R − iλ1I tanh(θ1 − δ1)] e−i (ζ1− ϕ1), θ1 ∼ O (1), t → + ∞ , ⎪ |λ1 |2 H1 H2 ⎩
(33)
⎧ w0 + 4 h1 sech2 (θ1 + δ1), θ1 ∼ O (1), t → − ∞ , ⎪ H1 H2 w→ 4 h1 ⎨ + w sech2 (θ1 − δ1), θ1 ∼ O (1), t → + ∞ , ⎪ 0 H 1 H2 ⎩
(34)
where
r1 =
4λ12I [λ12I + (λ1R − λ2R )2 + λ 22I ]2 − 16λ14I λ 22I ,
ϕ1 = arg(Λ2 + i Λ1),
δ1 = ln
ϕˆ1 = arg(−Λ2 − i Λ1) = ± π + ϕ1,
H2 , H1 ϕ˜1 = arg(−Λ2 + i Λ1) = ± π − ϕ1.
(35) (36)
From the asymptotic state analysis of two-soliton solution, it can be seen that the elastic interaction property between the two solitons is preserved. Following the same procedure, the asymptotic state solutions for θ2 ∼ O(1) when t→ ± ∞ are
r2 ⎧ sech(θ2 − δ1) e−i (ζ2− ϕ2), t → − ∞ , ⎪ H1 H2 q→ ⎨ r2 sech(θ2 + δ1) e−i (ζ2+ ϕ2), t → + ∞ , ⎪ H1 H2 ⎩
(37)
r ω ˆ ⎧ 22 0 sech(θ2 − δ1)[λ2R + iλ2I tanh(θ2 − δ1)] e−i (ζ1− ϕ2), t → − ∞ , ⎪ |λ2 | H1 H2 p→ r2 ω0 ˜ ⎨ sech(θ2 + δ1)[λ2R − iλ2I tanh(θ2 + δ1)] e−i (ζ2− ϕ2), t → + ∞ , ⎪ |λ2 |2 H1 H2 ⎩
(38)
⎧ w0 + 4 h2 sech2 (θ2 − δ1), t → − ∞ , ⎪ H1 H2 w→ 4 h2 ⎨ 2 ⎪ w0 + H H sech (θ2 + δ1), t → + ∞ , 1 2 ⎩
(39)
where 4
Optik - International Journal for Light and Electron Optics 200 (2020) 163348
X.-H. Meng, et al.
Fig. 1. Plots of the one-soliton solution in (a1)–(c1) with parameters: ω0 = −0.5, λ1 = 0.3 + 0.3 i.
r2 = 4λ42 [λ 22 + (λ1 − λ3 )2 + λ42 ]2 − 16λ 22 λ44 ,
ϕ2 = arg(Λ3 + i Λ1),
ϕˆ2 = arg(−Λ3 − i Λ1) = ± π + ϕ2 ,
(40)
ϕ˜2 = arg(−Λ3 + i Λ1) = ± π − ϕ2 .
(41)
4. Dynamical properties of the soliton solutions In this paper, the multi-soliton solutions in determinant form for the TSRS equations have been obtained using the Darboux transformation. In addition, the asymptotic state solutions for the two solitons have been analyzed, which indicates the interaction between the two solitons is elastic when |λ1|2 ≠ |λ2|2. In the following, by choosing different spectral parameters and ω0, abundant soliton evolution situations will be discussed. The positive and negative parameter ω0 corresponds to the anti-bell and bell shape soliton of w component, and the absolute value of ω0 affects the soliton width of the three components. The bigger the absolute value of ω0 is, the smaller the soliton width is. For different spectral parameter condition, two different soliton structures of p component will appear. As seen from Figs. 1(b1) and 2(b2) , when the absolute value of the real part of the spectral parameter is the same with the absolute value of the imaginary part, i.e. |λ1R| = |λ1I|, the soliton |p| is with a flat top. A two-hump soliton |p| will be shown as Figs. 3(b3) and 4(b4) when |λ1R| < |λ1I|. While a normal bell shape soliton |p| appears when the absolute value of the real part of the spectral parameter is larger than the absolute value of its imaginary part, i.e. |λ1R| > |λ1I|. Abundant two-soliton structures can be found for different spectral parameters. For two pure imaginary spectral parameters, seen from Fig. 5(a5)–(c5) , the two solitons interact with each other, and separate remaining their own original shape, while in Fig. 6(a6)–(c6) , two parallel soliton propagation phenomena can be seen. For two pure imaginary spectral parameters λ1 and λ2, the absolute value of the real part of the spectral parameter is less than the absolute value of its imaginary part, the two solitons of |p| are both two-hump solitons. When λ1 = −λ 2* with nonzero real and imaginary parts, the bound two soliton interaction occurs seen from Fig. 7(a7)–(c7) , and solitons propagation of the three components present periodical feature. Fig. 8(a8)–(c8) demonstrates the two soliton interaction with one pure imaginary spectral parameter and one complex spectral parameter with the same real and imaginary parts. From the figures, it can be seen that the |p| soliton consists of one flat-top soliton and one two-hump soliton. 5. Conclusions Based on the Darboux transformation, the multi-soliton solutions for the TSRS equations have been obtained in determinant forms. For different spectral parameters, some interesting soliton interaction phenomena have been demonstrated. More soliton structures can be illustrated for corresponding spectral parameters and the occurrence condition of different soliton structures can be analysed from the analytic solutions. The results obtained in this paper are hoped to be applicable in a variety of physical problems including the simulated Raman scattering.
Fig. 2. The corresponding section plots of the one-soliton solution in Fig. 1 at t = 0. 5
Optik - International Journal for Light and Electron Optics 200 (2020) 163348
X.-H. Meng, et al.
Fig. 3. Plots of the one-soliton solution in (a3)–(c3) with parameters: ω0 = 1, λ1 = 0.2 + 0.5 i.
Fig. 4. The corresponding section plots of the one-soliton solution in Fig. 3 at t = 0.
Fig. 5. Plots of the two-soliton solution with parameters: ω0 = 1, λ1 = 0.3 i, λ2 = 0.5 i.
Conflict of interest There is no potential conflict of interest.
Acknowledgements This work has been supported by the National Natural Science Foundation of China under Grant Nos. 11401031 and 11971067, Beijing Natural Science Foundation under Grant No. 1182009, the Beijing Great Wall Talents Cultivation Program under Grant No. CIT&TCD20180325, Qin Xin Talents Cultivation Program of Beijing Information Science and Technology University (QXTCPB201704 and QXTCP A201702Scientific Research Common Program of Beijing Municipal Commission of Education under Grant No. KM201911232011 and Modern measurement and control technology ministry of education key laboratory funding, the project of promoting the development of University connotation of Beijing Information Science and Technology University (5121911032, 5211910965, 5121911031 and 5221835201). 6
Optik - International Journal for Light and Electron Optics 200 (2020) 163348
X.-H. Meng, et al.
Fig. 6. Plots of the two-soliton solution with parameters: ω0 = 1, λ1 = −0.4999 i, λ2 = 0.5 i.
Fig. 7. Plots of the two-soliton solution with parameters: ω0 = 1, λ1 = 0.3 + 0.4 i, λ2 = −0.3 + 0.4 i.
Fig. 8. Plots of the two-soliton solution with parameters: ω0 = 1, λ1 = 0.5 + 0.5 i, λ2 = 0.4 i.
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