Darboux transformation, bright and dark-bright solitons of an N -coupled high-order nonlinear Schr¨odinger system in an optical ﬁber

In this paper, an N -coupled high-order nonlinear Schr¨odinger system, which describes the properties of the ultrashort optical pulses in an optical ﬁber, is investigated with the help of Darboux transformation (DT) method and asymptotic analysis. Starting from the given (2 N +1)th-order Lax pair, we construct a new form of DT (complex eigenfunctions of Lax pair involved) to derive the formulas of the n th-iterated solutions, where n and N are the positive integers. On the zero background, the ﬁrst- and second-order solitons are obtained and analysed through the asymptotic analysis. Multi-parameter adjustment is proceeded since there are 3 N +4 real parameters in the second-order solitons. We ﬁnd that under certain conditions each of the two interaction patterns (elastic, inelastic) holds in the second-order soliton. On the plane wave background, the ﬁrst-order bright and dark-bright solitons are obtained. Soliton velocities, amplitudes, widths and characteristic lines of the ﬁrst-order bright and dark-bright solitons are presented and analysed.

An N -coupled high-order nonlinear Schrödinger (HNLS) system has been proposed to describe the properties of the ultrashort optical pulses in an optical fiber [10][11][12][13][14][15]: |u n | 2 ) T = 0 (j = 1, 2, ..., N ), (1) where u j = u j (Z, T ) denotes the amplitude of the jth component of a slowly varying field, i = √ −1, n and N are the positive integers, the subscripts Z and T denote the normalized space and time derivatives, respectively. u jT T T represents the third-order dispersion, 6u jT N n=1 |u n | 2 represents the self-steepening and 3u j ( N n=1 |u n | 2 ) T represents the stimulated Raman scattering effects [11].
Via the variable transformations [15,16] q j (x, t) = u j (Z, T )e − i System (1) has been reduced to the N -coupled complex modified Korteweg-de Vries-type system of the form [17,18]: For System (3), Riemann-Hilbert problem [17] and Hirota bilinear technique [18] have been studied. Special cases of System (1) have been seen as follows: -When N = 1, System (1) has been reduced to the Sasa-Satsuma equation whose Bäcklund transformation [19], infinitely-many conservation laws [19] and Darboux dressing transformation [20] have been investigated. Sasa-Satsuma equation has been used to describe the development of the complex scalar field and evolutions of the short optical pulses [19].
However, to our knowledge, for System (1), a DT involving certain complex eigenfunctions, which is different from DT (20) in Ref. [13], has not been studied. Meanwhile, two solitons on the zero background and solitons on the plane wave background have not been investigated. In Sect. 2, such a DT of System (1) will be constructed, and the nth-iterated solutions of System (1) will be derived through that DT. In Sect. 3, through that DT and certain zero seed solutions, the first-and second-order soliton of System (1) will be derived on the zero background, and asymptotic analysis will be used to discuss the interaction between the two solitons. In Sect. 4, the first-order bright and dark-bright soliton of System (1) will be obtained on the plane wave background. In Sect. 5, the conclusions will be drawn.

DT of System (1)
Since System (1) and System (3) can be converted to each other through Variable Transformation (2), we can get the solutions of System (3) first, then perform Variable Transformation (2) and get the solutions of System (1).
Through DT (5), Lax Pair (4) is transformed into a new one possessing the same matrix form as followed: where U [1] , V [1] have the same forms with U, V , respectively, except that the r j , s j in the matrices U, V are replaced with r [1] j , s [1] j in the matrices U [1] , V [1] . Combining Eqs. (4), (5) and (6), we get We expand Eqs. (7) and let the coefficients of each subterm of λ be zero, then we get T are other eigenfunctions of Lax Pair (4) with λ = λ * 1 . We choose with where (H [1] ) −1 is the inverse of H [1] . Different from Ref. [13], we only require that the first eigenfunction φ 1 (λ 1 ) is real but the other eigenfunctions can be complex.
Eq. (9a) gives the relations between (q 1 , q 2 , ..., q N ) T and (q N ) T as followed: Now we have already constructed a DT of System (3), which involves complex eigenfunctions of the (2N +1)th-order Lax Pair (4). Eq. (13) gives the relation between the initial and new solutions of System (3). Eq. (5) gives the relation between the initial eigenfunction Φ and new eigenfunction Φ [1] of Lax Pair (4).
Iterating the above steps n times, the nth-iterated solutions of System (3) are presented as: Via Variable Transformation (2), we can get the nth-iterated solutions of System (1) where q j [x, t]. Eqs. (14)- (16) infer that we can obtain the nth-iterated solutions of System (1) with the initial seed solutions. There are three points that need to be explained: (i) The whole iterative process does not contain differential and integral operations. We only need to solve the linear Lax Pair (4) and obtain new solutions through repetitive matrix operations. (ii) The expressions of the nth-iterated solutions contain certain free parameters, e.g., spectral parameter λ k and integral constants in the seed solutions. (iii) DT (5) demands that λ k must be a pure imaginary number and φ 1 (λ k ) must be real. (1) (1) In order to derive solitons of System (3), we take seed solutions as q 1 = q 2 = ... = q N = 0 and let λ 1 = iη 1 (η 1 ∈ R and η 1 = 0). Substituting the seed solutions into Lax Pair (4), we can get eigenfunctions as followed:
Taking the Eigenfunctions (17) into Eq. (13), we can get the first-order soliton of System (3) Via Variable Transformation (2), we can get the first-order soliton of System (1) From Solutions (19), we can obtain the following soliton properties: (a) Soliton Velocities: V (u , (e) Characteristic Lines: (8η 3 1 + 1 6 η 1 )Z − 2η 1 T − δ 1 + δ = 0. We find that all the one solitons enjoy the same velocity, width and characteristic line. The soliton width is in inverse proportion to η 1 while the velocity decreases as |η 1 | increases, and the velocity is always positive. Different one solitons have different amplitudes due to the different δ j+1,R . The total energy keeps constant and it's independent with δ 1 . The first-order soliton for System (1) with N = 3 is plotted in Figs. 1 in which the surface and shape of the one soliton are presented.
Via Variable Transformation (2), we can get the second-order soliton of System (1) as u [2] j (Z, T ) = q
We find that δ hC do not affect the modulus value of u [2] j in Solution (22). Then, we let δ hC equal to 0 without affecting our following analysis.

Figs. 2. Elastic interaction between the two solitons via Solutions
Under the conditions: ξ N +1,R = 1 or δ 1 = ξ 1 = δ hR = ξ hR = 0, the interaction is elastic. It means that before and after interaction, the two solitons only have a phase shift. Their velocities, amplitudes, shapes, and widths do not change at all.
As shown in Figs. 2, the elastic interaction under the two conditions above occurs between the second-order solitons for System (1) with N = 3.
The values of η 1 and η 2 affect the angle between the two solitons in elastic interaction. The angle increases as the value of ||η 1 | − |η 2 || increases. The angle will reduce to zero but not equal to zero (means that the two solitons are nearly parallel showed in Figs. 5 when η 1 η 2 ∼ −1.

(B) Elastic interaction case II:
Under the conditions: −δ 1 = δ hR and −ξ 1 = ξ hR , the interaction is elastic and Solutions (22) reduce to u [2] where Motivated by Ref [26], without loss of generality, we assume that 0 < η 1 < η 2 . For Solutions (23), if ω 1 is fixed, ω 2 can be expressed as When T → −∞, we have e −ω 2 → 0; When T → +∞, we have e ω 2 → 0. We can get If ω 2 is fixed, ω 1 can be expressed as When T → −∞, we have e ω 1 → 0; When T → +∞, we have e −ω 1 → 0. So we can get All the two solitons own the same shape, amplitude and velocity. Results of the asymptotic analysis show that after the elastic interaction, the properties changed are merely the characteristic lines.
(C) Inelastic interaction: Under the conditions: ξ N +1,R = 0, 1, the interaction is inelastic. We take the second-order soliton for System (1) with N = 3 as an example, inelastic interaction occurs as seen in Figs. 3 and 4. When k 1, the solitons S1 in the u 1 , u 2 components are suppressed and the soliton S1 in the u 3 component is enhanced; While the solitons S2 in the u 1 , u 2 components are enhanced and the soliton S2 in the u 3 component is suppressed.

Bright and dark-bright solitons of System (1) on the plane wave background
To get the bright and dark-bright solitons, we choose the following plane wave seed solutions of System (3): where a j and b j are real constants, b j are all distinct, g is a positive integer, a j ≥ 0. According to Solutions (14), we firstly need to solve Lax Pair (4) with Seed Solutions (24) via the gauge transformation: Φ = DΨ, D = diag (1, e −iθ 1 , e iθ 1 , e −iθ 2 , e iθ 2 , ..., e −iθ N , e iθ N ).
Ψ satisfies the following Lax pair: In order to convert K to a diagonal matrix, we proceed the following matrix decomposition . µ 1 , µ 2 , ..., µ 2N +1 are 2N + 1 distinct roots of the (2N + 1)th-order characteristic equation in the variable µ: where det denotes the determinant of a matrix. We simplify the (2N +1)th-order characteristic equation, then get Eigenfunctions (25) can be given as Φ = DM QL with Q = diag(e F (µ 1 ) , e F (µ 2 ) , ..., e F (µ 2N +1 ) ), where l m (m = 1, 2, ..., 2N + 1) are non-zero real constants. After calculation we get: To make sure that φ 1 is a real function and Φ satisfies DT (5), l m and µ m need be real while λ need be a pure imaginary. It also means that all the solutions of the (2N +1)-order Characteristic Equation (26) need be real.

The first-order bright soliton of System (1)
Without loss of generality, we suppose that λ = iξ, µ 1 , µ 2 > 0 and µ 1 = µ 2 . Under this constraints l 3 = l 4 = ... = l 2N +1 = 0, we substitute Eq. (27) into Solutions (16) and use the Variable Transformation (2), then we get the first-order bright soliton , ). (28) We find that the breathers don't appear under this condition. q j are always a series of bright solitons whether l m are equal to zero or not.

The first-order dark-bright soliton of System (1)
In order to get the dark-bright soliton of System (1), without loss of generality, motivated by Ref [27], we assume that there are N 1 dark solitons (N 1 is a positive integer and 1 ≤ N 1 < N ) and N − N 1 bright solitons. For this we let a f = 0 (f = 1, 2, ..., N 1 ), means that u f (x, t) are dark solitons and the rest (N − N 1 ) components u e (x, t) (e = N 1 + 1, N 1 + 2, ..., N ) are bright solitons.
To illustrate these dark-bright solitons, we choose the first-order dark-dark-bright solitons for System (1) with N = 3 as an example as showed in Figs. 6 (corresponding to three identical characteristic roots).

Conclusion
In this paper, an N -coupled HNLS system (N is a positive integer), i.e., System (1) which describes the properties of the ultrashort optical pulses in an optical fiber, has been investigated. System (1) has been converted into System (3) through Variable Transformation (2). Based on the (2N +1)th-order Lax Pair (4) of System (3), DT (5) for System (1) has been constructed, which demands that the first element φ 1 in the eigenfunction Φ is real and complex spectral parameter λ 1 = iη 1 is a pure imaginary number. By virtue of DT (5), the nth-iterated solutions of System (1) (n is a positive integer) have been constructed as Solution (16).
Analytic forms of the first-order soliton on the zero background have been got as Solutions (19). Properties of the solitons have been investigated: Solitons enjoy the same velocity, width and characteristic line; Widths of the solitons are inverse to η 1 while their velocities decrease as η 1 increase; Velocities of the solitons are always positive; Amplitudes of the solitons in each component are proportional to e δ j+1,R (δ j+1,R are the integral constants), as is shown in Figs. 1.
Analytic forms of the second-order soliton on the zero background have been got as Solutions (21) and (22) with 3N + 4 real parameters. Effects of those parameters on the two solitons have been analysed by virtue of the asymptotic analysis: -Elastic interaction case I and case II have been shown in Figs. 2. Via the asymptotic analysis, we have found that before and after the interaction, the two solitons only have a phase shift.
-Inelastic interaction between the two solitons have been shown in Figs. 3 and 4. We have analyzed the amplitude changes of solitons S1 and S2 in each component, which imply the increase or decrease of soliton energy.
-Nearly parallel two solitons have been shown in Figs. 5. As long as the spectral parameters λ 1 = −λ 2 , the two solitons will eventually meet and interact elastically or inelastically.
The first-order bright and dark-bright soliton for System (1) has been derived on the plane wave background as Solutions (28) and (29), respectively. Properties of these solitons have been investigated: Velocities and widths of bright solitons in Solutions (28) are related to the two solutions (µ 1 and µ 2 ) of (26); All the dark solitons and bright solitons in Solutions (29) move at the same velocity as shown in Figs. 6.