Introduction
The concept of integrable couplings has been introduced and various integrable couplings have been studied systematically. This originated from an investigation on centerless Virasoro symmetry algebras of integrable systems [4, 10] . Generally, for a given integrable system u t = K(u) , an integrable coupling is a triangular integrable system
Introduction
where the potentials u and v can be either scalar functions or vector functions of the dependent variables x and t, and ∂S/∂[u] ̸ = 0, such that the whole system is not separated. At the beginning, integrable couplings were constructed by using perturbations [4, 5, 10] , or enlarging spectral problems and Lie algebras [6, 21] . Later, it was found that integrable couplings have a close connection with semidirect sums of Lie algebras, and a new method to construct integrable couplings was presented [13, 14] . The existing approaches are just specific examples of the semidirect sum method, or in other words, applications of semidirect sums. On one hand, an integrable coupling enlarges the original integrable system, which adds new equations, and thus it may be more complicated than the original system but it enriches the considered integrable system. On the other hand, for some concrete equations, based on the physical problems that the original equations describe, the supplementary equations can help us understand the underlying physical problems better.
Introduction
The theory of Hamiltonian structures is important to study integrable systems. A continuous Hamiltonian system of a PDE reads
Introduction
where J is a Hamiltonian operator, H = ∫ Hdx is the Hamiltonian functional, and δ δu is the variational derivative with respect to u. Magri [16] introduced the celebrated bi-Hamiltonian formulation, which paves the way for generating infinitely many symmetries and conservation laws. The bi-Hamiltonian formulation often implies the Liouville integrability of PDEs. There are several ways to construct Hamiltonian structures of integrable systems. A very powerful and simple technique by using the trace identity was proposed by Tu [17, 18] , which heavily relies on the Killing form of Lie algebras. Because the Killing form is nondegenerate if and only if the Lie algebra is semisimple, the trace identity method is valid only for classical integrable systems that are generated from semisimple Lie algebras. For integrable couplings, the underlying Lie algebras are nonsemisimple, and the trace identity method is not valid anymore. To solve this problem, a variational identity method was proposed in [9] , which aims to construct Hamiltonian structures for integrable couplings. It is a generalization of the trace identity method.
Introduction
Many examples of hierarchies of integrable couplings have been presented for classical soliton hierarchies, including the KdV hierarchy, the MKdV hierarchy, the Toda hierarchy, the Glachette-Johnson hierarchy, the Jaulent-Miodek hierarchy, the Benjamin-Ono hierarchy, the Tu hierarchy, the AKNS hierarchy, and the KaupNewell hierarchy [8, 15, 19] . There are also studies on biintegrable couplings [7] and tri-integrable couplings [12] , which aim to classify integrable couplings from the Lie algebra point of view. The supertrace identity is presented to study superintegrable systems [11] . The Darboux transformation for integrable couplings was also presented [20] .
Introduction
In this paper, by using the zero curvature equation of an enlarged AKNS spectral problem, we shall derive a new hierarchy of integrable couplings and then explore its Hamiltonian structure by the variational identity. The paper is structured as follows. In Section 2, we shall construct a novel hierarchy of integrable couplings from a specific semidirect sum of Lie algebra associated with sl (2) . In Section 3, by applying the variational identity, we shall construct a bi-Hamiltonian structure for the resulting hierarchy of coupling systems, thereby showing their Liouville integrability. A few remarks will be given at the end of the paper.
Novel integrable couplings from an enlarged AKNS spectral problem
The AKNS soliton hierarchy [1] is a general source to construct nonlinear integrable equations. We shall construct a novel hierarchy of integrable couplings for the AKNS soliton hierarchy. The key point is to enlarge the spatial spectral problem based on a non-semisimple matrix loop algebraḡ. The enlarged spectral problem has an additional matrix block depending on the spectral parameter λ, and such an enlarged spatial spectral problem is a first try to generate hierarchies of integrable couplings.
Novel integrable couplings from an enlarged AKNS spectral problem
Letsl(2) denote the loop algebra associated with the special linear algebra sl (2) :
Novel integrable couplings from an enlarged AKNS spectral problem
In order to construct integrable couplings, we enlarge the simple Lie algebrasl(2) to a non-semisimple Lie algebraḡ. Consider the Lie algebra consisting of 2 × 2 block matrices:
Novel integrable couplings from an enlarged AKNS spectral problem
It is direct to see thatḡ is a semidirect sum of the simple Lie algebra g and the solvable Lie algebra g c , i.e.
Novel integrable couplings from an enlarged AKNS spectral problem
The semidirect sum means [g,
Novel integrable couplings from an enlarged AKNS spectral problem
denotes the Lie bracket ofḡ . Therefore, g c is an ideal Lie subalgebra ofḡ .
Novel integrable couplings from an enlarged AKNS spectral problem
Then we consider an enlarged special spectral problem:
Novel integrable couplings from an enlarged AKNS spectral problem
where λ is the spectral parameter,φ is a four-dimensional column vector of eigenfunctions, andū = (p, q, r, s)
Novel integrable couplings from an enlarged AKNS spectral problem
T denotes the potential. The complement part U a in this enlarged Lax operator depends explicitly on the spectral parameter λ , which is the first try to generate integrable couplings.
Novel integrable couplings from an enlarged AKNS spectral problem
The stationary zero curvature equation
Novel integrable couplings from an enlarged AKNS spectral problem
induces
Novel integrable couplings from an enlarged AKNS spectral problem
i.e.
Novel integrable couplings from an enlarged AKNS spectral problem
(2.10)
Novel integrable couplings from an enlarged AKNS spectral problem
substituting them into (2.10), and comparing the coefficients of λ , we obtain the recursion relations:
Novel integrable couplings from an enlarged AKNS spectral problem
Let us take the initial values
Novel integrable couplings from an enlarged AKNS spectral problem
where α and β are arbitrary constants, and we assume that α ̸ = 0 . In addition, choose the constants of integration to be zero:
Novel integrable couplings from an enlarged AKNS spectral problem
i can be computed from (2.13) and (2.14), respectively. Now let us introduce the temporal spectral problems
Novel integrable couplings from an enlarged AKNS spectral problem
with 19) where m ≥ 0 and P + denotes the polynomial part of P in λ. Then the compatibility conditionŪ t −V
Novel integrable couplings from an enlarged AKNS spectral problem
Equation (2.20) generates the original AKNS soliton system, while Equation (2.21) results in a supplementary soliton subsystem. Noting the recursion relations (2.13) and (2.14), we obtain a hierarchy of AKNS integrable couplings: 
Novel integrable couplings from an enlarged AKNS spectral problem
where the recursion operatorΦ is defined bȳ
Novel integrable couplings from an enlarged AKNS spectral problem
The first few equations are computed as follows:
Novel integrable couplings from an enlarged AKNS spectral problem
If we denote
Novel integrable couplings from an enlarged AKNS spectral problem
then the above integrable coupling hierarchy (2.24) can be written as
Novel integrable couplings from an enlarged AKNS spectral problem
and the recursion operatorΦ is determined by the recursion relation
Novel integrable couplings from an enlarged AKNS spectral problem
A lengthy computation shows thatΦ is hereditary [2, 3] , i.e. it satisfies
Novel integrable couplings from an enlarged AKNS spectral problem
for all vector fieldsK andS . We point out that a case with an additional matrix block
Novel integrable couplings from an enlarged AKNS spectral problem
was investigated in [9] , where v i , i = 1, 2, 3 are all new dependent variables. Here in our situation, the submatrix operators Φ and Φ c are the same as the ones in [9] , but the recursion operatorΦ in (2.25) is different from the one in [9] , and we introduce a new soliton hierarchy of AKNS integrable couplings.
Hamiltonian structures of the resulting AKNS integrable couplings
To construct Hamiltonian structures of the resulting coupling systems, we need to compute a nondegenerate, symmetric, and ad-invariant bilinear form on the enlarged matrix Lie algebra that we adopted above:
Hamiltonian structures of the resulting AKNS integrable couplings
This Lie algebra has been used in [9] , and we would like to recall the construction procedure of its bilinear forms made in [9] as follows.
Hamiltonian structures of the resulting AKNS integrable couplings
There are four steps to construct the required bilinear forms on the above Lie algebraḡ : (1) Construct an isomorphism between the loop algebraḡ and a Lie algebra in vector form; The whole procedure of constructing the required bilinear forms is as follows. First, we transform the Lie algebraḡ into a Lie algebra in vector form. Let us define a mapping:
Hamiltonian structures of the resulting AKNS integrable couplings
This mapping induces a Lie algebraic structure on R 6 , which is isomorphic to the original matrix Lie algebrā g . The corresponding commutator on R 6 should satisfy:
Hamiltonian structures of the resulting AKNS integrable couplings
where R(b) can be computed as
Hamiltonian structures of the resulting AKNS integrable couplings
Then a bilinear form on R 6 can be determined by
Hamiltonian structures of the resulting AKNS integrable couplings
where F is a constant matrix. The nondegenerate property requires det F ̸ = 0 , the symmetric property ⟨a, b⟩ R 6 = ⟨b, a⟩ R 6 requires F T = F , and the ad-invariance property
Hamiltonian structures of the resulting AKNS integrable couplings
Solving this system, we obtain
Hamiltonian structures of the resulting AKNS integrable couplings
Thus, the required bilinear forms on the original Lie algebraḡ read
Hamiltonian structures of the resulting AKNS integrable couplings
Obviously, such bilinear forms are symmetric and ad-invariant:
Hamiltonian structures of the resulting AKNS integrable couplings
where A, B, C ∈ḡ , and they are nondegenerate if and only if η 2 ̸ = 0 .
Hamiltonian structures of the resulting AKNS integrable couplings
Recall that the continuous variational identity [9] reads δ δū ) . Otherwise, we need to construct a bilinear form on the underlying Lie algebra. The Lie algebra we considered here has the same form as the one in [9] , and thus the required bilinear form reads as in [9] :
Hamiltonian structures of the resulting AKNS integrable couplings
where η 1 , η 2 are two arbitrary constants, and
Hamiltonian structures of the resulting AKNS integrable couplings
It is symmetric and ad-invariant: 13) and the nondegenerate property requires that η 2 ̸ = 0 .
Hamiltonian structures of the resulting AKNS integrable couplings
In order to apply the variational identity (3.11), we compute that 
Hamiltonian structures of the resulting AKNS integrable couplings
Taking m = 1 yields the constant γ = 0. Therefore, we have δ δū
Hamiltonian structures of the resulting AKNS integrable couplings
where m ≥ 1. Noting the notation (2.30), the right-hand side of (3.17) can be denoted by
Hamiltonian structures of the resulting AKNS integrable couplings
where
Hamiltonian structures of the resulting AKNS integrable couplings
It follows now that the enlarged AKNS coupling systems in (2.24) possess the following Hamiltonian structures: 20) where the Hamiltonian operatorJ is given bȳ 21) and the Hamiltonian functionals are given bȳ
Hamiltonian structures of the resulting AKNS integrable couplings
Notice that the recursion operatorΦ is hereditary, and we can have a Hamiltonian pair consisting ofJ and M =ΦJ . Therefore, it follows that the Hamiltonian couplings (3.20) possess a bi-Hamiltonian structure, and furthermore, they are all Liouville integrable. In particular, they have infinitely many commuting symmetries {K m |m ≥ 0} and conserved functionals {H m |m ≥ 0} , which form two abelian Lie algebras:
Hamiltonian structures of the resulting AKNS integrable couplings
where m 1 , m 2 ≥ 0.
Concluding remarks
In this paper, we constructed a novel hierarchy of integrable couplings for the AKNS soliton equations through a different choice of selecting block matrices, which leads to a novel non-semisimple loop algebra. We showed that the resulting coupling systems have bi-Hamiltonian structures with a recursion operator, which leads to Liouville integrability. The theory of integrable couplings brings other interesting results, such as Lax pairs of block forms and Lax pairs with several spectral parameters, and helps us work towards a complete classification of multiple component integrable systems.
Concluding remarks
The presented enlarged spatial spectral problem in (2.4) and (2.5) associated with the non-semisimple matrix loop algebra defined in (2.2) is of a completely new type. Such a novelty brings complexities and difficulties in computing soliton hierarchies, but new features on integrable couplings. In particular, the dependence of the spectral parameter λ in the additional matrix block can guarantee the effectiveness of applying the Darboux transformation method to construct soliton solutions of integrable couplings. It is expected that more applications of such a new class of enlarged spectral problems can be explored from different points of view. Moreover, an additional interesting question for us is whether any integrable coupling must be linear, i.e. the added subsystem in the coupling system must be linear with the added variable.

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