On Construction of a Super Hierarchy of the Wadati-Konno- Ichikawa Equation

The super extensions of the standard integrable systems in two-dimensional spacetime have been investigated for the recent several decades. Many classical integrable equations have been extended to be the super completely integrable equations, such as the super Korteweg-de Vries (KdV) equation [1–3], super AKNS [4–7], super Kadomtsev-Petviashvili (KP) [8], super Kaup-Newell (KN) [9], super Camassa-Holm (CH) [10], super vector nonlinear Schrödinger equations [11], super Heisenberg [12], and so on [13–20]. The Wadati-Konno-Ichikawa (WKI) equation, proposed in [21], can be written in the form

The Wadati-Konno-Ichikawa (WKI) equation, proposed in [21], can be written in the form which can be used to describe the nonlinear oscillation of elastic beam under tension. In this paper, we propose a super WKI hierarchy associated with a 3 × 3 matrix spectral problem, in which the first nontrivial member takes the following form: which is the well-known WKI equation (1) as α = 0 and β = 0. The outline of this paper is as follows. In Section 2, we introduce a 3 × 3 matrix spectral problem with two commuting potentials u and v, and two anticommuting potentials α and β. This spectral problem is an extension of the spectral problems associated with the WKI equation. From this spectral problem, a hierarchy of the super WKI equations are proposed with the aid of the zero-curvature equation. In Section 3, the super bi-Hamiltonian structures of the super WKI hierarchy are constructed by using the super trace identity [22][23][24][25][26]. In Section 4, we derive infinite conservation laws of the super WKI equation by resorting to the spectral parameter expansions. For the applied and analytic aspects on conservation laws, one can refer to [27][28][29][30]. We can refer to the two most recent results on the mixed method for the calculation of conservation laws studied in [29,30].

Super WKI Equations
In this section, a hierarchy of super WKI equations will be obtained. We first introduce a 3 × 3 matrix spectral problem: where u, v, λ, ϕ 1 , and ϕ 2 are the commuting variables, which can be indicated by the degree (mod 2) p as pðuÞ = pðvÞ = pðλÞ = pðϕ 1 Þ = pðϕ 2 Þ = 0; α, β, and ϕ 3 are the anticommuting variables which can be indicated by p as pðαÞ = pðβÞ = pðϕ 3 Þ = 1. In order to derive the hierarchy of super nonlinear evolution equations associated with the spectral problem (3), we need to solve the stationary zero-curvature equation: where where each entry V ij = V ij ðA, B, C, ρ, δÞ is a function of A, B, C, ρ, and δ: with pðAÞ = pðBÞ = pðCÞ = 0, pðρÞ = pðδÞ = 1. Substituting (6) into (5), we have Advances in Mathematical Physics The functions A, B, C, ρ, and δ are expanded as the following Laurent series in λ: Substituting (8) into (7), we can get the Lenard recursion equation as follows: where and K and J are two operators defined by with and ∂ = ∂ x .
To find a general representation of the solution for (9), we present a Lenard recursion equation as follows: with condition to identify constants of integration as zero when acting with operator J −1 upon Kg j . This means that g j is uniquely determined by the recursion equation (13).

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Operating with ðJ −1 KÞ j upon G −1 = c 0 g −1 , we get the general solution of (9).
where c 0 , c 1 , ⋯, c j+1 are constants of integration and g j = where each entry V ðnÞ ij = V ij ðA ðnÞ , B ðnÞ , C ðnÞ , ρ ðnÞ , δ ðnÞ Þ in the matrix V ðnÞ is a polynomial of eigenparameter λ with where A j , B j , C j , ρ j , and δ j are determined by (15). Then, the compatibility condition of (3) and (16) yields the zerocurvature equation U t n − V ðnÞ x + ½U, V ðnÞ = 0, which is equivalent to the hierarchy of the super WKI equations.

Super Bi-Hamiltonian Structures
In this section, the super bi-Hamiltonian structures of equation (19) will be established by using the super trace identity as follows [23][24][25][26][27]: where γ is a constant to be determined and U 0 = ðu, v, α, βÞ T . It is easy to observe that Substituting (22) and (8) into (21), we arrive at where Assuming that I = 1 + uM + αΓ, F = ðV 13 ΓÞ/λ, (35) can be rewritten as I t = F x , which is the right form of conservation laws. We expand I and F as series in powers of λ with the coefficients which are called conserved densities and currents, respectively where c 0 is a integration constant of (15). The first members of conserved densities and currents are as follows: The recursion relations of I j and F j (j ≥ 1) are as follows: where M j and Γ j can be computed by (33).

Data Availability
The data used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that they have no conflicts of interest.
Advances in Mathematical Physics