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

Authors: Xuemei Li; Lutong Li
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

In this paper, a super Wadati-Konno-Ichikawa (WKI) hierarchy associated with a3×3 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.


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## Body

## 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
(1)iut=u1+uvxx,ivt=−v1+uvxx,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:
(2)ut=i−u1+uvx+2uα1+uvβ1+uvx−u1+uvαβ1+uvx+2ααx1+uv3x,vt=iv1+uvx+2vβ1+uvα1+uvx+v1+uvαβ1+uvx−2ββx1+uv3x,αt=i−21+uvα1+uvx−2u1+uvβ1+uvx+uα1+uvv1+uvx−β1+uvu1+uvx−3uαββx−2−uvααxβ1+uv5x,βt=i21+uvβ1+uvx−2v1+uvα1+uvx−vβ1+uvu1+uvx−α1+uvv1+uvx+3vααxβ+2−uvαββx1+uv5x,which is the well-known WKI equation (1) as α=0 and β=0.The outline of this paper is as follows. In Section2, 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 [22–26]. 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 a3×3 matrix spectral problem:
(3)ϕx=Uϕ,ϕ=ϕ1ϕ2ϕ3,U=λλuλαλv−λλβ−λβλα0,where u, v, λ, ϕ1, and ϕ2 are the commuting variables, which can be indicated by the degree (mod 2) p as pu=pv=pλ=pϕ1=pϕ2=0; α, β, and ϕ3 are the anticommuting variables which can be indicated by p as pα=pβ=pϕ3=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:
(4)Vx=U,V,V=Vij3×3,where pV11=pV12=pV21=pV22=pV33=0; pV13=pV23=pV31=pV32=1. We note that equation (4) is equivalent to
(5)V11x=−λvV12+λuV21+λαV31−λβV13,V12x=2λV12+λuV22−V11+λαV13+V32,V13x=λV13+λuV23+λαV33−V11−λβV12,V21x=−2λV21+λvV11−V22+λβV31−V23,V22x=λvV12−λuV21+λαV23+λβV32,V23x=−λV23+λvV13−λαV21+λβV33−V22,V31x=−λV31−λvV32+λαV21+λβV33−V11,V32x=λV32−λuV31+λαV22−V33−λβV12,V33x=λαV23+V31+λβV32−V13,where each entry Vij=VijA,B,C,ρ,δ is a function of A, B, C, ρ, and δ:
(6)V11=−A,V12=B,V13=δ,V21=C,V22=A,V23=ρ,V31=−ρ,V32=δ,V33=0,with pA=pB=pC=0, pρ=pδ=1. Substituting (6) into (5), we have
(7)Ax=λvB−λuC+λαρ+λβδ,Bx=2λB+2λuA+2λαδ,Cx=−2λC−2λvA−2λβρ,δx=λδ+λuρ+λαA−λβB,ρx=−λρ+λvδ−λαC−λβA.The functionsA,B,C,ρ, and δ are expanded as the following Laurent series in λ:
(8)A=∑j=0∞Ajλ−j,B=∑j=0∞Bjλ−j,C=∑j=0∞Cjλ−j,ρ=∑j=0∞ρjλ−j,δ=∑j=0∞δjλ−j.Substituting (8) into (7), we can get the Lenard recursion equation as follows:
(9)KGj−1=JGj,JG−1=0,j≥0,where
(10)Gj−1=Cj,Bj,−2ρj,2δj,AjT,and K and J are two operators defined by
(11)K=12Kij5×5,J=12Jij5×5,with
(12)Kij=0∂1+uv−αβ1+uv∂∂uα1+uv∂−∂α1+uv∂0−K120−∂β1+uv∂−∂vβ1+uv∂0−K13−K23−∂u1+uv+2αβ1+uv2∂∂1+uv+1−uvαβ1+uv2∂0−K14−K24K34∂v1+uv+2αβ1+uv2∂0−u∂−v∂−α∂−β∂2∂,Jij=∂u∂−1u∂2∂+∂u∂−1v∂∂u∂−1α∂∂u∂−1β∂02∂+∂v∂−1u∂∂v∂−1v∂∂v∂−1α∂∂v∂−1β∂0∂α∂−1u∂∂α∂−1v∂∂α∂−1α∂∂+∂α∂−1β∂0∂β∂−1u∂∂β∂−1v∂−∂+∂β∂−1α∂∂β∂−1β∂0−u∂−v∂−α∂−β∂2∂,and ∂=∂x.To find a general representation of the solution for (9), we present a Lenard recursion equation as follows:
(13)Kgj−1=Jgj,j≥0,with condition to identify constants of integration as zero when acting with operator J−1 upon Kgj. This means that gj is uniquely determined by the recursion equation (13). We choose kerJ=c0g−1=c01+uv−3/21+uv+αβ−v,−u,2β,−2α,1T. Through long calculations, we have
(14)g0=vx21+uv3+3vαβx−αxβ−3v2ααx−2−uvββx21+uv5−ux21+uv3+3uαβx−αxβ+3u2ββx+2−uvααx21+uv5vxα+2vαx−2βx1+uv3−3uvxvα−β21+uv5−uxβ−2uβx−2αx1+uv3+3uvxuβ+α21+uv5uvx−uxv41+uv3−2−uvαβx−αxβ−3vααx−uββx21+uv5.Operating withJ−1Kj upon G−1=c0g−1, we get the general solution of (9).
(15)Gj=c0gj+c1gj−1+⋯+cj+1g−1,j≥−1,where c0,c1,⋯,cj+1 are constants of integration and gj=gj1,gj2,gj3,gj4,gj5T.Letϕ satisfy the spectral problem (3) and the following auxiliary problem
(16)ϕtn=Vnϕ,where each entry Vijn=VijAn,Bn,Cn,ρn,δn in the matrix Vn is a polynomial of eigenparameter λ with
(17)An=∑j=0n−1Ajλn+1−j,Bn=∑j=0n−1Bjλn+1−j+λBn+uAn,Cn=∑j=0n−1Cjλn+1−j+λCn+vAn,ρn=∑j=0n−1ρjλn+1−j+λρn+βAn,δn=∑j=0n−1δjλn+1−j+λδn+αAn,where Aj,Bj,Cj,ρj, and δj are determined by (15). Then, the compatibility condition of (3) and (16) yields the zero-curvature equation Utn−Vxn+U,Vn=0, which is equivalent to the hierarchy of the super WKI equations.
(18)utn,vtn,αtn,βtnT=PKGn−2=PJGn−1,n≥1,where P is a projective map η1,η2,η3,η4,η5T⟶η1,η2,η3,η4T. This can be transformed into
(19)utn,vtn,αtn,βtnT=c0Xn−1+c1Xn−2+⋯+cn−1X0,n≥1,with Xj=PKgj−1=PJgj. The first nontrivial member in the hierarchy (19) is as follows:
(20)ut1=c02−u1+uvx+2uα1+uvβ1+uvx−u1+uvαβ1+uvx+2ααx1+uv3x,vt1=c02v1+uvx+2vβ1+uvα1+uvx+v1+uvαβ1+uvx−2ββx1+uv3x,αt1=−c0221+uvα1+uvx+2u1+uvβ1+uvx−uα1+uvv1+uvx+β1+uvu1+uvx+3uαββx−2−uvααxβ1+uv5x,βt1=−c02−21+uvβ1+uvx+2v1+uvα1+uvx+vβ1+uvu1+uvx+α1+uvv1+uvx−3vααxβ+2−uvαββx1+uv5x,which is reduced to the famous WKI equation (1) (see [21, 31]) as t1=t, α=0, β=0, c0=2i or the super WKI equation (2) as t1=t, c0=2i.


## 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]:
(21)δδU0∫strV∂U∂λdx=λ−γ∂∂λλγstr∂U∂U0V,where γ is a constant to be determined and U0=u,v,α,βT. It is easy to observe that
(22)strV∂U∂λ=−2A+vB+uC−2αρ+2βδ,str∂U∂uV=λC,str∂U∂vV=λB,str∂U∂αV=−2λρ,str∂U∂βV=2λδ.Substituting (22) and (8) into (21), we arrive at
(23)δδU0∫−2A+vB+uC−2αρ+2βδdx=λ−γ∂∂λλγ+1C,B,−2ρ,2δT,δδU0∫−2Aj+vBj+uCj−2αρj+2βδjdx=γ+1−jCj,Bj,−2ρj,2δjT,j≥0.Through direct calculations, we find thatγ=k as ck≠0, cll=0,l≠kj=0, 0≤k≤j. Using (23) and noticing (15), we have
(24)δδU0Hj=Cj,Bj,−2ρj,2δjT,j≥0,where
(25)H0=−2c0∫1+uv−αβ1+uvdx,H1=c0∫uxv−uvx2uv1+uv+αxβ−αβx+vααx−uββx1+uv3dx−2c1∫1+uv−αβ1+uvdx,Hj=∑k=0j−2ckk+1−j∫u−∂−1u∂gj−k1+v−∂−1v∂gj−k2+α−∂−1α∂gj−k3+β−∂−1β∂gj−k4dx+cj−1∫uxv−uvx2uv1+uv+αxβ−αβx+vααx−uββx1+uv3dx−2cj∫1+uv−αβ1+uvdx,j≥2.From (24), we obtain the desired Hamiltonian form of (19) as follows:
(26)utn,vtn,αtn,βtnT=KδδU0Hn−1=JδδU0Hn,where K and J are two super-Hamiltonian operators defined by
(27)K=12Kij4×4,J=12Jij4×4,with Kij and Jij given by (11). Especially, the super WKI equation (20) can be written as follows:
(28)ut1,vt1,αt1,βt1T=KδδU0H0=JδδU0H1.


## 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
(29)M=ϕ2ϕ1,Γ=ϕ3ϕ1,where ϕ1, ϕ2, and ϕ3 satisfy (3) and (16) with n=1. Noticing that ϕ32=0, we get from (3) that
(30)Mx=λv−2M+βΓ−uM2−αMΓ,Γx=λ−β+αM−Γ−uMΓ.We expandM,Γ in powers of λ−1 as follows:
(31)M=∑j=0∞Mjλ−j,Γ=∑j=0∞Γjλ−j,where pMj=0,pΓj=1. Substituting (31) into (30) and comparing the coefficients of the same powers of λ, we obtain
(32)M0=1u−1±11+uv,Γ0=1uα∓uβ+α1+uv,and a recursion formula for Mj and Γj
(33)Mj+1=∓121+uvMjx−Γ0Γjx+u∑k=1jMkMj−k+1−uΓ0−α∑k=1jMkΓj−k+1,Γj+1=∓11+uvΓjx−αMj+1+u∑k=0jMj−k+1Γk,j≥0.Since(34)∂∂tϕ1xϕ1=∂∂xϕ1tϕ1,we can derive the conservation law of (20) as follows:
(35)∂∂t1+uM+αΓλ=∂∂xV111+V121M+V131Γ,where
(36)V111=−c0g−15λ2,V121=c0−ug−15λ+g02+ug05λ,V131=c0−αg−15λ+g042+αg05λ.Assuming thatI=1+uM+αΓ, F=V111+V121M+V131Γ/λ, (35) can be rewritten as It=Fx, 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
(37)I=∑j=0∞Ijλ−j,F=∓c0λ+∑j=0∞Fjλ−j,where c0 is a integration constant of (15). The first members of conserved densities and currents are as follows:
(38)I0=±1+uv−αβ1+uv,F0=±c04uvx−uxv1+uv−uvx−uxvαβ−6vααx+6uββx−22−uvαxβ−αβx1+uv2.The recursion relations ofIj and Fj(j≥1) are as follows:
(39)Fj=c0−ug−15Mj+1+g02+ug05Mj−αg−15Γj+1+g042+αg05Γj,Ij=uMj+αΓj,j≥1,where Mj and Γj can be computed by (33).



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*Source: 6838731-2020-02-07.xml*