Generalized Matrix Exponential Solutions to the AKNS Hierarchy

1School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China 2Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA 3College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China 4Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China 5College of Mathematics and Physics, Shanghai University of Electric Power, Shanghai 200090, China 6Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa


Introduction
Many nonlinear models are studied and shown to possess hierarchies, and recursion operators play a crucial role in constructing hierarchies of soliton equations [1]. Associated with the variational derivative, recursion operators have been developed to formulate the Hamiltonian structures proving the integrability of soliton hierarchies [2,3]. Recursion operators also have a tight correlation with one-soliton solutions. For example, the Korteweg-de Vries (KdV) hierarchy = , = 0, 1, 2, . . . , possesses the following one-soliton solution: where = 2 + 4 + 2 −1 is the recursion operator of the KdV hierarchy, is a constant, and = / , −1 = −1 = 1. Notice that the dispersion relation of (2) is linked closely with the order of the recursion operator .
Since the discovery of scattering behavior of solitons [4] and the IST [5], solitons have received much attention. The IST has been also well developed and widely used to solve nonlinear equations [1,6]. It can be used to solve not only normal soliton equations but also unusual soliton equations such as equations with self-consistent sources [7], nonisospectral equations [8,9], and equations with steplike finite-gap backgrounds [10] and on quasi-periodic backgrounds [11]. Recently, Ablowitz and Musslimani developed the IST for the integrable nonlocal nonlinear Schrödinger (NLS) equation [12].
The Sylvester equation is one of the most well-known matrix equations. It appears frequently in many areas of applied mathematics and plays a central role, in particular, in systems and control theory, signal processing, filtering, model reduction, image restoration, and so on. In recent years, it has been used to solve soliton equations [13,14] successfully. The method based on the 2

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Sylvester equation is also known as Cauchy matrix approach [14][15][16][17][18], which is actually a by-product of direct linearization approach first proposed by Fokas and Ablowitz in 1981 [15] and developed to discrete integrable systems by Nijhoff et al. in early of 1980s [16].
In the process of solving soliton equations, it can yield fantastic results by using matrices properly. Successful examples are the Wronskian technique and the IST. Ma et al. had introduced the matrix element in Wronskian determinants when they used the Wronskian technique to solve soliton equations [19][20][21][22][23]. They obtained various kinds of solutions such as soliton, rational, Matveev, and complexiton solutions. In 2006, Aktosun and Van Der Mee proposed a modified inverse scattering transformation (MIST) [24]. They expressed the scattering data of spectral problems by three matrices, , , , and proved that the matrices , , satisfy the Sylvester equation. The advantage of the method is that it can get more kinds of solutions and its process is simpler than the traditional IST [24][25][26][27].
In this paper, we would like to consider the AKNS hierarchy Many integrable systems can be reduced from it such as the modified KdV, sine-Gordon, nonlinear Schrödinger, and nonlocal nonlinear Schrödinger system. There have had already a lot of researches on AKNS hierarchy for it is a representative integrable hierarchy [21,22,28]. In this paper, we will construct the soliton, complexiton, and Matveev solutions to the first nonlinear equation of (5). We will show that the linear relation (3) exists not only in one-soliton solutions but also in multisoliton, complexiton, and Matveev solutions. The paper is organized as follows: In Section 2, we will review the recovered potentials of the AKNS spectral problem by IST. In Section 3, we will obtain the coupled Sylvester equations and generalized matrix exponential solutions to the AKNS equation. In Section 4, some different types of explicit solutions will be constructed. In Section 5, generalized matrix exponential solutions will be given to the AKNS hierarchy. The relationship between the recursion operator and the solutions of the AKNS hierarchy will be discussed also. We conclude the paper in Section 6.

Preparation
To make the paper self-contained, we first briefly recall the Lax integrability of the isospectral AKNS hierarchy and its Gel'fand-Levitan-Marchenko (GLM) equations.
It is well known that the AKNS hierarchy has the following Lax pairs [29]: , is a spectral parameter, and = ( , ), = ( , ) are potential functions. We assume that ( , ) and ( , ) are smooth functions of variables and , and their derivatives of any order with respect to vanish rapidly as → ∞. The compatibility condition, zero curvature equation with boundary conditions can yield the isospectral AKNS hierarchy (5). The first two nonlinear equations in the AKNS hierarchy are Next we mainly follow the notions and results given in [1,6].
the spectral problem (7a) has a group of Jost solutions ( , ), ( , ), ( , ), and ( , ) which are bounded for all values of and also enjoy the following asymptotic behaviors:
In order for (14a) and (14b) to be valid, it is necessary that Definition 2. If and are single roots of ( ) and ( ), respectively, there exist and such that and are named the normalization constants for the eigenfunctions ( , ) and ( , ). Accordingly, and ( , ) are named the normalization eigenfunctions.
to be the scattering data for the spectral problem (7a).

Lemma 4. Given the scattering data for the spectral problem (7a) and
( ) = ( ) + ( ) , one has the GLM equations for the AKNS hierarchy

Generalized Matrix Exponential Solutions to the AKNS Equation
In this section, we will get generalized matrix exponential solutions to the AKNS equation (9a) via MIST. From the previous section, we know that the potentials of spectral problem (7a) can be recovered by (15). For convenience, we denote 1 ( , ) and 2 ( , ) by ( , ) and ( , ), respectively, in this and the following sections.
Proof. Rewrite (19a) and (19b) in their component forms as one has the following coupled Sylvester relations: Proof. We only prove the first relation, and the second relation can be proved similarly: For convenience, we set where , are × matrices, , , , , , , , are matrices defined in Lemma 6, and they enjoy the relations (24). Setting if Γ and Γ are nondegenerate matrices, we have the following theorem.
Proof. Let ( ) and ( ) be zero and ( ) and ( ) be matrix exponential forms in (18a); that is, Suppose that the time evolution of ( ) and that of ( ) are That is, In the light of we have Similarly, we can arrive at Finally, we find that the potentials of the AKNS spectral problem (7a) can be recovered as by taking advantage of (15).
Next we will get solutions to the AKNS equation (9a) from the recovered potentials (28).
Although Proposition 8 is simple, we will use it many times in the following part of this paper. Proof. We only prove the first equation of (9a), and the second equation can be proved similarly. On the one hand and on the other hand Through proper simplification, we have

Exact Solutions to the AKNS Equation
In this section, we will construct different kinds of explicit solutions to the AKNS equation (9a) by taking different kinds of , , and , , .

The Same Linear Relation That a General Equation and Its Solutions Share
In this section, we will construct a new general equation related to the AKNS hierarchy and get its generalized matrix exponential solutions. Furthermore, we find that the recursion operator and the solutions of the AKNS hierarchy have the same linear relation.
We consider a new general equation: where is a polynomial operator on . Let we have the following Lemma.
Lemma 10. If = and = , one obtains Proof. We only prove the first equation. For we have From (28), we know that the conclusion is right.
Proof. We use the mathematical induction to prove the theorem. When = 1, we have and (56) is established. Supposing that we obtain that is, ) .
Through tedious calculation, we arrive at By Lemma 10, we know that the first equality in (60) is right. Its second equality can be proved analogously.
That is, / = = . In the same way, we also can obtain = when (2 ) = (2 ) . Thus we have The corollary means that the operator and the solutions of the AKNS hierarchy enjoy the same linear structure.

Conclusions
To sum up, we have solved the AKNS hierarchy by the MIST. The MIST can determine solutions more directly and generate more diverse solutions than the traditional IST. Actually, such solutions can be also obtained by applying the Wronskian technique [19][20][21][22][23]. The arbitrariness of the matrices involved leads to the diversity of exact solutions.
A general integrable equation of the AKNS hierarchy was constructed and its matrix exponential solutions were obtained. The ways to generate the equation and its matrix exponential solutions are the same. They share the same linear algebraic structure, not only in the case of one-soliton solutions but also in the case of other interesting solutions.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.