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

Abstract

In this paper, a super Wadati-Konno-Ichikawa (WKI) hierarchy associated with a matrix spectral problem is derived with the help of the zero-curvature equation. We obtain the super bi-Hamiltonian structures by using of the super trace identity. Infinitely, many conserved laws of the super WKI equation are constructed by using spectral parameter expansions.

1. Introduction

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 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 matrix spectral problem, in which the first nontrivial member takes the following form: which is the well-known WKI equation (1) as and .

The outline of this paper is as follows. In Section 2, we introduce a matrix spectral problem with two commuting potentials and , and two anticommuting potentials and . 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 [22–26]. 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 [27–30]. We can refer to the two most recent results on the mixed method for the calculation of conservation laws studied in [29, 30].

2. Super WKI Equations

In this section, a hierarchy of super WKI equations will be obtained. We first introduce a matrix spectral problem: where , , , , and are the commuting variables, which can be indicated by the degree (mod 2) as ; , , and are the anticommuting variables which can be indicated by as . 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 ; . We note that equation (4) is equivalent to where each entry is a function of , , , , and : with , . Substituting (6) into (5), we have

The functions , and are expanded as the following Laurent series in :

Substituting (8) into (7), we can get the Lenard recursion equation as follows: where and and are two operators defined by with and .

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 upon . This means that is uniquely determined by the recursion equation (13). We choose Through long calculations, we have

Operating with upon , we get the general solution of (9). where are constants of integration and

Let satisfy the spectral problem (3) and the following auxiliary problem where each entry in the matrix is a polynomial of eigenparameter with where , and are determined by (15). Then, the compatibility condition of (3) and (16) yields the zero-curvature equation , which is equivalent to the hierarchy of the super WKI equations. where is a projective map . This can be transformed into with . The first nontrivial member in the hierarchy (19) is as follows: which is reduced to the famous WKI equation (1) (see [21, 31]) as , , , or the super WKI equation (2) as , .

3. 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 [23–27]: where is a constant to be determined and . It is easy to observe that

Substituting (22) and (8) into (21), we arrive at

Through direct calculations, we find that as , , . Using (23) and noticing (15), we have where

From (24), we obtain the desired Hamiltonian form of (19) as follows: where and are two super-Hamiltonian operators defined by with and given by (11). Especially, the super WKI equation (20) can be written as follows:

4. Conservation Laws

In this section, infinitely, many conservation laws of the super WKI equation (20) will be constructed. First, let us introduce the variables where , , and satisfy (3) and (16) with . Noticing that , we get from (3) that

We expand in powers of as follows: where . Substituting (31) into (30) and comparing the coefficients of the same powers of , we obtain and a recursion formula for and

Since we can derive the conservation law of (20) as follows: where

Assuming that , , (35) can be rewritten as , which is the right form of conservation laws. We expand and as series in powers of with the coefficients which are called conserved densities and currents, respectively where is a integration constant of (15). The first members of conserved densities and currents are as follows:

The recursion relations of and () are as follows: where and 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.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 11871440).