Riemann–Hilbert approach and N-soliton solution for an eighth-order nonlinear Schrödinger equation in an optical fiber

This paper aims to present an application of the Riemann–Hilbert approach to treat higher-order nonlinear differential equation that is an eighth-order nonlinear Schrödinger equation arising in an optical fiber. Starting from the spectral analysis of the Lax pair, a matrix Riemann–Hilbert problem is formulated strictly. Then, by solving the obtained Riemann–Hilbert problem under the reflectionless case, N-soliton solution is generated for the eighth-order nonlinear Schrödinger equation. Finally, the localized structures and dynamic behaviors of one- and two-soliton solutions are illustrated by some figures.


Introduction
In this paper, we investigate in detail an eighth-order nonlinear Schrödinger (NLS) equation which is used for describing the propagation of ultrashort nonlinear pulses. 1,2 It can be generated from truncating the infinite hierarchy of nonlinear Schrödinger equations 3 that is used to investigate the higher-order dispersive effects and nonlinearity. Here q(x, t) denotes a normalized complex amplitude of the optical pulse envelope. And the subscripts of q(x, t) mean the partial derivatives with respect to the scaled spatial coordinate x and time coordinate t correspondingly. Each coefficient A j (2 ≤ j ≤ 8) is an arbitrary real number, and Here the superscript * represents complex conjugate.
As a matter of fact, Equation (1) covers many nonlinear differential equations of important significance, some of which are listed as follows: (1) is reduced to the fundamental nonlinear Schrödinger equation describing the propagation of the picosecond pulses in an optical fiber.
(ii) For the case of A 2 = 1 2 and A 4 = A 5 = A 6 = A 7 = A 8 = 0, Equation (1) is reduced to the Hirota equation 4,5 describing the third-order dispersion and time-delay correction to the cubic nonlinearity in ocean waves.
(iii) For the case of A 2 = 1 2 and A 5 = A 6 = A 7 = A 8 = 0, Equation (1) becomes a fourth-order dispersive NLS equation 6,7 describing the ultrashort optical-pulse propagation in a long-distance, high-speed optical fiber transmission system.
(iv) For the case of A 2 = 1 2 and A 6 = A 7 = A 8 = 0, Equation (1) becomes a fifth-order NLS equation 8 describing the attosecond pulses in an optical fiber.
By now, there have been plenty of researches on Equation (1). For instance, the interactions among multiple solitons were under study, 1 and oscillations in the interaction zones were observed systematically. As a result, it was found that the oscillations in the solitonic interaction zones possess different forms with different spectral parameters of Equation (1) and so forth. In a follow-up study, 2 the Lax pair and infinitely-many conservation laws were derived via symbolic computation, which verifies the integrability of Equation (1). Moreover, the one-, two-and threesoliton solutions were explored as well by means of the Darboux transformation.
The principal aim of this study is to determine multi-soliton solutions for the eighth-order NLS equation (1) with the aid of the Riemann-Hilbert approach. 9−20 This paper is divided into five sections. In second section, we recall the Lax pair associated with Equation (1) and convert it into a desired form. In third section, we carry out the spectral analysis, from which a Riemann-Hilbert problem is set up on the real line. In fourth section, the construction of multi-soliton solutions for Equation (1) is detailedly discussed in the framework of the Riemann-Hilbert problem under the reflectionless case. A brief conclusion is given in the final section.

Lax pair
Upon the Ablowitz-Kaup-Newell-Segur formalism, the eighth-order NLS equation (1) admits a 2 × 2 Lax pair 2 where Ψ = (Ψ 1 , Ψ 2 ) T is a vector eigenfunction, Ψ 1 and Ψ 2 are the complex functions of x and t, the symbol T signifies transpose of the vector, and ς is a isospectral parameter. Furthermore, Let us now rewrite the Lax pair (2) in a more convenient form where andâ l mean q(x, t) and its derivative terms appeared in a l (2 ≤ l ≤ 6).

Riemann-Hilbert problem
In this section, we focus on putting forward a Riemann-Hilbert problem for the eighth-order NLS equation (1). Now we assume that the potential function q(x, t) in the Lax pair (3) decays to zero sufficiently fast as x → ±∞. It can be known from (3) that when x → ±∞, which motivates us to introduce the variable transformation Upon this transformation, the Lax pair (3) can be changed into the desired form where [·, ·] is the matrix commutator and U 1 = iQ. From (4), we find that tr(U 1 ) = tr(Q 1 ) = 0.
In the direct scattering process, we will concentrate on the spectral problem (4a), and the tdependence will be suppressed. We first introduce two matrix Jost solutions µ ± of (4a) expressed as a collection of columns meeting the asymptotic conditions Here the subscripts of µ indicated refer to which end of the x-axis the boundary conditions are required for, and I stands for the identity matrix of size 2. Actually, the solutions µ ± are uniquely determined by the integral equations of Volterra type After direct analysis on Equations (7) we can see that [µ − ] 1 , [µ + ] 2 are analytic for ς ∈ C − and continuous for ς ∈ C − ∪R, while [µ + ] 1 , [µ − ] 2 are analytic for ς ∈ C + and continuous for ς ∈ C + ∪R, where C − and C + are respectively the lower and upper half ς-plane: Next we set out to study the properties of µ ± . In fact, it can be shown from tr(U 1 ) = 0 that the determinants of µ ± are independent of the variable x. Evaluating det µ − at x = −∞ and det µ + at x = +∞, we get det µ ± = 1 for ς ∈ R. In addition, µ − E and µ + E are both fundamental solutions of (3a), where E = e iςσx , they are linearly dependent Here S(ς) = (s kj ) 2×2 is called the scattering matrix and det S(ς) = 1. Furthermore, we find from the properties of µ ± that s 11 allows analytic extension to C − and s 22 analytically extends to C + .
A Riemann-Hilbert problem desired is closely associated with two matrix functions: one is analytic in C + and the other is analytic in C − . In consideration of the analytic properties of µ ± , we set defining in C + , be an analytic function of ς. And then, P 1 can be expanded into the asymptotic series at large-ς Inserting expansion (10) into the spectral problem (4a) and equating terms with same powers of ς, we obtain i σ, P For establishing a Riemann-Hilbert problem, the analytic counterpart of P 1 in C − is still needed to be given. Noting that the adjoint scattering equation of (4a) reads as and the inverse matrices of µ ± meet this adjoint equation. Then we express the inverse matrices of µ ± as a collection of rows which obey the boundary conditions µ −1 ± → I as x → ±∞. It is easy to know from (8) that where R(ς) = (r kj ) 2×2 = S −1 (ς). Thus, the matrix function P 2 which is analytic for ς ∈ C − is constructed as Analogous to P 1 , the very large-ς asymptotic behavior of P 2 turns out to be P 2 → I as ς ∈ C − → ∞.
Carrying (5) into Equation (8) gives rise to Hence, P 1 is of the form On the other hand, via substituting (12) into Equation (13), we get As a consequence, P 2 is written as With two matrix functions P 1 and P 2 which are analytic in C + and C − respectively in hand, we are in a position to deduce a Riemann-Hilbert problem for the eighth-order NLS equation (1).
After denoting that the limit of P 1 is P + as ς ∈ C + → R and the limit of P 2 is P − as ς ∈ C − → R, a Riemann-Hilbert problem can be given below with its canonical normalization conditions as and r 21 s 12 + r 22 s 22 = 1.

N -soliton solution
Having described a Riemann-Hilbert problem for Equation (1), we now turn to construct its multisoliton solutions. To achieve the goal, we first need to solve the Riemann-Hilbert problem (15) under the assumption of irregularity, which signifies that both det P 1 and det P 2 possess some zeros in the analytic domains of their own. From the definitions of P 1 and P 2 , we have which means that det P 1 and det P 2 have the same zeros as s 22 and r 22 respectively, and r 22 = (S −1 ) 22 = s 11 .
With above analysis, it is now necessary to reveal the characteristic feature of zeros. It can be noticed that the potential matrix Q has the symmetry property Q † = Q, upon which we deduce Here the subscript † stands for the Hermitian of a matrix. In order to facilitate discussion, we introduce two special matrices H 1 = diag(1, 0) and H 2 = diag(0, 1), and express (9) and (14) in terms of A direct computation of the Hermitian of expression (17a), using the relation (16), generates that and the involution property of scattering matrix S † (ς * ) = S −1 (ς), which leads to This equality implies that each zero ±ς k of s 22 results in each zero ±ς * k of r 22 correspondingly.
Therefore, our assumption is that det P 1 has simple zeros {ς j ∈ C + , 1 ≤ j ≤ N } and det P 2 has simple zeros {ς j ∈ C − , 1 ≤ j ≤ N }, whereς l = ς * l , 1 ≤ l ≤ N. The full set of the discrete scattering data is composed of these zeros and the nonzero column vectors υ j and row vectorsυ j , which satisfy the following equations Taking the Hermitian of Equation (20a) and using (18) as well as comparing with Equation (20b), we find that the eigenvectors fulfill the relation Differentiating Equation (20a) about x and t and taking advantage of the Lax pair (4), we arrive at Here υ j,0 , 1 ≤ j ≤ N, are complex constant vectors. Making use of the relation (21), we havê However, in order to derive soliton solutions of the eighth-order NLS equation (1), we investigate the Riemann-Hilbert problem (15) corresponding to the reflectionless case, i.e., s 12 = 0.
thus the solution to the problem (15) can be determined by where M −1 kj denotes the (k, j)-entry of M −1 . From expression (22a), it can be seen that In what follows, we shall retrieve the potential function q(x, t) based on the scattering data.
Expanding P 1 (ς) at large-ς as and carrying this expansion into (4a) give rise to Consequently, the potential function is reconstructed as with P (1) 1 21 being the (2,1)-entry of P 1 . To conclude, setting the nonzero vectors υ k,0 = (α k , β k ) T and Im(ς k ) > 0, the general N -soliton solution for the eighth-order NLS equation (1) is written as where The bright one-and two-soliton solutions will be our main concern in the rest of this section.

Conclusion
In this investigation, the aim was to explore multi-soliton solutions for an eighth-order nonlinear Schrödinger equation arising in an optical fiber. The method we resort to was the Riemann-Hilbert approach which is based on a Riemann-Hilbert problem. Therefore, we first described a Riemann-Hilbert problem via analyzing the spectral problem of the Lax pair. After solving the obtained Riemann-Hilbert problem corresponding to the reflectionless case, we finally generated the expression of general N -soliton solution to the eighth-order nonlinear Schrödinger equation. In addition, the localized structures and dynamic behaviors of bright one-and two-soliton solutions  were shown graphically via suitable choices of the involved parameters.

Funding
This work was supported by the National Natural Science Foundation of China (Grant Nos. 61072147 and 11271008).