The integrability, equivalence and solutions of two kinds of integrable deformed fourth-order matrix NLS equations

Based on the higher-order restricted flows, the first type of integrable deformed fourth-order matrix NLS equations, that is, the fourth-order matrix NLS equations with self-consistent sources (FMNLSSCS), is derived. By virtue of the ∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{\partial }}$$\end{document}-dressing method, the second type of integrable deformed fourth-order matrix NLS equations called the fourth-order matrix NLS–Maxwell–Bloch system (FMNLS-MB) is presented. We prove the equivalence of the FMNLSSCS and the FMNLS-MB successfully. Furthermore, N-soliton solutions are explicitly obtained by means of the Cauchy matrix method starting from corresponding Sylvester equation.


Introduction
In recent years, the matrix nonlinear Schrödinger (NLS) equations have been studied in many theoretical works [1][2][3][4][5]. The classic matrix NLS equation is defined as [4,5] iQ t + Q x x + 2QQ † Q = 0, where Q = q 1 q 2 −q 2 q 1 is a 2 × 2 matrix valued potential function, q 1 and q 2 denote slowly varying complex amplitudes, " †" denotes the conjugate and transpose, subscripts x and t denote partial derivatives with respect to the space and time coordinates, respectively.
The above system has been turned out to be of relevance for the description of F = 1 spinor Bose-Einstein condensates and the dynamics of orthogonally polarized nonlinear waves in the isotropic medium [6][7][8][9][10][11].
With the development of nonlinear system theory, more and more studies on nonlinear systems involve higher-order effects, such as self-steepening effects, higher-order dispersion and self-frequency effects [12][13][14]. Sometimes, the dispersion and dissipation on the fiber materials cause the deviations in soliton width amplitude that obtained from the NLS equation. Higher-order effects can usually reduce this deviation. It is revealed that the higher-order effects modify the solitary characteristics in communicating fibers [15]. In addition, the presence of higher-order terms can affect the velocity of solitons and the oscillation of soliton intensity [16]. Therefore, it is important to consider the higher-order effects of nonlinear systems in realistic physical systems. When considering the propagation of the ultra-short pulses in optical fiber, the classic matrix NLS equation obviously cannot be well characterized. Therefore, in this paper, we consider the matrix version of the higher-order NLS system [17], which reads as where α is the coefficient of higher-order dispersion and the nonlinear terms. The system (2) plays key roles in explaining the propagation of the ultra-short optical pulse in optical fibers with the higher-order nonlinear effects [18]. For this reason, the system (2) may has broad application prospects in the field of longdistance, high-speed optical fiber transmission. In addition, the nonlinear spin excitations in one-dimensional isotropic biquadratic Heisenberg ferromagnetic spin with the octupole-dipole interaction [19,20] can be governed by the higher-order NLS equation. Therefore, the matrix version of the higher-order NLS system has potential application prospects in more complex Heisenberg ferromagnetic spin phenomenon. The integrable deformation of integrable system has attracted extensive attention in mathematical and physical fields. One kind of the integrable deformation is the so-called soliton equation with self-consistent sources (SCS) [21][22][23][24][25][26], which consists of the soliton equation with additional terms by coupling the corresponding eigenvalue problems. This kind of integrable deformation can explain varieties of phenomena of great physical significance, including but not limited to hydrodynamics, plasma physics and solid-state physics. For instance, the NLS equation with SCS describes soliton propagation in a medium with both resonant and nonresonant nonlinearities and the nonlinear interaction of high-frequency electrostatic wave with ion acoustic waves in plasma [25,26]. It should be pointed out that the existence of SCS enriches the nonlinear dynamics of solitons. For example, the insertion of a source may cause changes in soliton velocity [27]. At the same time, the soliton equation with SCS also possesses good properties [28,29]. There are usually two types of solitons in optical fibers: NLS solitons and MB solitons. When these two kinds of solitons exist in the same fiber medium, like erbium-doped fibers, the resulting pulse phenomenon is more complex, so its propagation needs to be depicted by higher-order matrix NLS-MB system.
In order to further analyze the influence of higherorder effects on soliton dynamics, in this paper, we construct two kinds of integrable deformed fourthorder matrix NLS equations, that is, the fourth-order matrix NLS equations with self-consistent sources and the fourth-order matrix NLS-Maxwell-Bloch sys- where N = n 1 n 2 −n 2 n 1 , M = m 1 m 2 −m 2 m 1 , is a scalar function in the singular dispersion relation. Further, we proved that FMNLSSCS (3) and FMNLS-MB (4) are equivalent and proposed the explicit expression of the N-soliton solution via the Cauchy matrix method [30][31][32].
The paper is arranged as follows. In Sect. 2, based on the higher-order restricted flows and the∂-dressing method, the FMNLSSCS (3) and FMNLS-MB (4) are deduced and their equivalence are proved. In Sect. 3, the explicit expression of the N-soliton solutions is established. Finally, we give the conclusion in Sect. 4.

Two kinds of integrable deformed fourth-order matrix NLS equation
In this section, we will deduce the FMNLSSCS (3) and FMNLS-MB (4) based on the higher-order restricted flows and the∂-dressing method, respectively.

Higher-order restricted flow and FMNLSSCS
We start from the linear spectral problem [17] where Next, we let The adjoint representation V x = [U, V ] yields the recurrence relations with the initial values and Then, the compatibility condition gives rise to the fourth-order matrix NLS hierarchy The higher-order restricted flows of the integrable deformed fourth-order matrix hierarchy which consist of the equations obtained from conserved quantities H n and λ j are where λ j are distinct. According to [24], the integrable deformed fourth-order matrix NLS hierarchy is given by We concentrate on the case n = 2 and denote t ≡ t 2 ; the matrix form of equation (14) is exactly the FMNLSSCS (3).

The∂-dressing method and FMNLS-MB
Now we consider the 4 × 4 matrix∂-problem in the complex planē with a boundary condition (z,z, is a spectral transform matrix. Thus, the solution of (15) with the canonical normalization can be written as where C z is the Cauchy-Green integrable operator acting on the left [33][34][35] RC z = 1 2πi Formally, the solution of the∂-problem (15) can be given as For the convenience, we introduce a notation [36] f, g = 1 2πi f which possesses the prosperities In order to construct FMNLS-MB, we need to introduce the x, t dependence in the transformation matrix R. We let where (z) is a dispersion relation. Here we consider the case that (z) comprises both a polynomial part p (z) and a singular part s (z), that is, where ω(z) is a scalar function. Based on the basic properties of the Cauchy-Green operator [36,37] where f (z) ≡ f (z), I . Equations (15), (16) and (21) give rise to the Zakharov-Shabat spectral problem and the adjoined time evolution equation where Theorem Based on the Zakharov-Shabat spectral problem, the integrable deformed four-order matrix NLS hierarchy can be expressed as Proof In order to derive the hierarchy of the integrable deformed four-order matrix NLS equations, we introduce the following symmetry about the potential P with σ = I 2 0 2×2 0 2×2 −I 2 , which implies that P takes (24), we can find that ψ(z) has the symmetry Further, we introduce a 4 × 4 matrix U = ψ J ψ −1 , it is easy to verify by virtue of (24) and (26) and Let where U (D) stands for diagonal matrix and U (O) stands for non-diagonal matrix, then (30) gives rise to two equations According to the asymptotic condition ψ → I as x → ∞, we obtain from the first equation in (33) that which means the second equation in (33) can be rewritten as Next, we introduce a new recursion operator and then, (31) takes the form (37) and (30) just is the integrable deformed four-order matrix NLS hierarchy.

The equivalence of FMNLSSCS and FMNLS-MB
In this section, we will discuss the equivalence of the FMNLSSCS (1) and FMNLS-MB (2).
Based on the eigenvalue problem of the fourth-order matrix NLS equation, we let n 1 := n 1 (k) = φ 1 φ 2 −φ 3φ4 , and then, we get where z = z+z 2 ∈ R. At this time, the system of equations (41) and (43) is equivalent to the FMNLS-MB system (4).

Cauchy matrix method and soliton solutions
In this section, we use Cauchy matrix method to derive the soliton solutions of the FMNLSSCS.

Theorem 2 The integrable deformed fourth-order matrix NLS equations (3) have the following N-soliton solution
where Proof We begin with Sylvester equation and the dispersion relations where We suppose E(K) ∩ E(−K) = ∅, then according to Sylvester Theorem, (45) has a unique solution M for given (L, K, R, S) [38], where E(K) denotes eigenvalue sets of K.
Next, we introduce master function and auxiliary matrix functions According to (47), we knowS Making using of (45) and (46), we have where a = For the sake of the resulting equations, according to (47), we denote where (46), we can work out the evolutions of u (i) and S (i, j) and (53) In the following, we consider the case of i = j = 0 in (53). Making using of the evolution of S (i, j) and the relation S (i, j+n) = S (i+n, j) − n−1 l=0 S (n−1−l, j) S (i,l) and taking whose solution is given by and then (54) can be rewritten as (3).
Up to now, we have established the links between the Sylvester equation (45) 1, 2, . . . , N , and with Under the above case, (55) and (56) give out the expression of the N-soliton solution of the FMNLSSCS (1).

Two-soliton-like solutions
When taking N = 2 in (44) and denoting ρ (0) , the explicit two-soliton-like solutions can be obtained. Because the expression of the solution is so complex, we don't list it here. We focus on its dynamic behavior in the following. It can be found that the functions β 1 (t) and β 2 (t) affect the propagation trajectory of two-solitonlike solutions, and the spectral parameters k 1 and k 2 affect the type and interaction of two-soliton-like solutions. In the following, the parameters are taking as α = 0.25, c 1 = 1, c 2 = 2.
When the real and imaginary parts of the spectral parameters k 1 and k 2 are all different, we can obtain soliton-breather-like solutions as shown in Fig. 3. Figure 3 also illustrates that the trajectories of propagation for |q 1 |, |q 2 | have changed and their amplitudes have increased after the interaction.
When the spectral parameters k 1 and k 2 have the same imaginary part, two-soliton-like waves arrived. Figure 4 shows that the two waves in |q 1 | propagate forward keeping original direction and the amplitudes of them decrease after interaction, while the propagation direction of the two waves in |q 2 | changes after interaction. Based on the above analysis, we find the following results by comparing Fig. 3 and Fig. 4. When we take the same β i (t) (i = 1, 2), the propagation trajectories of |q 1 | and |q 2 | are the same. At this time, different k i (i = 1, 2) leads to the appearance of different types of two-soliton-like solutions and interactions, for example, the shape and deviation in the propagation direction of two waves before and after the collision. Different from Fig. 4, when β 1 (t) and β 2 (t) are trigonometric functions, the solutions are periodic. Figure 5 shows that two waves (|q 1 |, |q 2 |) propagate periodically over time when β 1 (t) = β 2 (t) = − sin t 4 .
From Fig. 3 to Fig. 5, it is straightforward to find that β 1 (t) and β 2 (t) influence the propagation trajectory of two-soliton-like solutions; k 1 and k 2 affect the type of two-soliton-like solutions and the deviation of the propagation direction of two waves before and after the collision. In addition, we can also find that the properties of |q 1 | and |q 2 | are not identical.

Conclusion
The study of higher-order integrable system is of great significance in both mathematics and physics. In this paper, two kinds of integrable deformed fourth-order matrix hierarchy including the FMNLSSCS and the FMNLS-MB are derived based on the higher-order Fig. 4 Interaction of two-soliton-like solutions at β 1 (t) = β 2 (t) = −1 and k 1 = 0.1 + 0.2i, k 2 = 0.3 + 0.2i: (a) the evolution of |q 1 |, (b) the evolution of |q 2 |, (c) the propagation of |φ 1 | at t = 1, t = 4, t = 6, (d) the propagation of |φ 2 | at t = 1, t = 4, t = 6, (e) the propagation of |φ 3 | at t = 1, t = 4, t = 6, (f) the propagation of |φ 4 | at t = 1, t = 4, t = 6 restricted flows and∂-dressing method, respectively. Starting from a 4 × 4 matrix∂-problem, we obtained the Lax pair by constructing a spectral transformation matrix. The equivalence of the FMNLSSCS and the FMNLS-MB is verified. Next, the general expres-sion of soliton solutions has been obtained through the Cauchy matrix method starting from corresponding Sylvester equation. Specially, the one-soliton solution, two-soliton-like solutions and breather solitonlike solutions are given. Data availability There are no data taken from outside sources.

Conflict of interest
The authors declare that they have no conflict of interest.