# A Soliton Hierarchy Associated with a Spectral Problem of 2nd Degree in a Spectral Parameter and Its Bi-Hamiltonian Structure

Authors: Yuqin Yao; Shoufeng Shen; Wen-Xiu Ma
Journal: Advances in Mathematical Physics
 (2016)

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## Abstract

Associated withso~(3,R), a new matrix spectral problem of 2nd degree in a spectral parameter is proposed and its corresponding soliton hierarchy is generated within the zero curvature formulation. Bi-Hamiltonian structures of the presented soliton hierarchy are furnished by using the trace identity, and thus, all presented equations possess infinitely commuting many symmetries and conservation laws, which implies their Liouville integrability.


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## Body

## 1. Introduction

Consider an evolution equation(1)ut=Ku=Ku,ux,…,where the field function u(x,t) is in a linear space S and K(u)=K(u,ux,…) is a suitable C∞ vector field. Our starting point is to determine isospectral deformations of a linear spatial spectral problem(2)ϕx=Uu,λϕ,where U is a square spectral matrix and λ is a spectral parameter [1–3]. With the linear spectral problem (2), let us associate an auxiliary temporal spectral problem (3)ϕt=Vu,λϕ,where V is a square matrix of the same order as U. The compatibility condition of (2) and (3) is the zero curvature equation (4)Ut-Vx+U,V=0.If an evolution equation (1) can be presented by such a zero curvature equation, we call it a soliton equation, and U and V a Lax pair of (1). In general, for a well-chosen spectral matrix U, we can obtain, from different particular choices of V, a hierarchy of soliton equations:(5)ut=JδHnδu,where J is a Hamiltonian operator and Hn’s are conserved functionals [1, 3, 4]. Many well-known examples of such soliton hierarchies are presented within the zero curvature formulation, which include the AKNS hierarchy [2, 5], the Kaup-Newell hierarchy [6, 7], the Wadati-Konno-Ichikawa hierarchy [8], the coupled Harry-Dym hierarchy [9], and the Dirac hierarchy [10].Recently, the three-dimensional real special orthogonal Lie algebraso(3,R) has been used to construct soliton hierarchies [11–13]. The Lie algebra so(3,R) consists of 3×3 trace-free, skew-symmetric matrices and has the basis(6)e1=00-1000100,e2=00000-1010,e3=0-10100000, with the commutator relations:(7)e1,e2=e3,e2,e3=e1,e3,e1=e2.The derived algebra [so(3,R),so(3,R)] is so(3,R) itself. This is one of only two three-dimensional real Lie algebras with a three-dimensional derived algebra. The other one is sl(2,R), which has been widely used in studying soliton equations [2, 6, 8, 12–15]. The corresponding matrix loop algebra is defined by (8)so~3,R=∑i≥0Miλn-i∣Mi∈so3,R,i≥0,n∈Z,that is, the space of all Laurent series in λ with a finite number of nonzero terms of positive powers of λ and coefficient matrices in so(3,R). Particular examples of this matrix loop algebra so~(3,R) contain the following linear combinations:(9)λme1+λne2+λle3with arbitrary integers m,n,l.Soliton hierarchies possess many nice properties, for instance, Lax representations or zero curvature representations, Hamiltonian structures, infinitely many conservation laws, and infinitely many symmetries. So it is interesting to search for new soliton hierarchies associated with a particular Lie algebra. In this paper, we would like to construct a spectral problem of 2nd degree in a spectral parameter, based on the matrix loop algebraso~(3,R). A hierarchy of commuting bi-Hamiltonian soliton equations will be generated from associated zero curvature equations. Bi-Hamiltonian structures will be furnished by using the trace identity, and thus, the resulting hierarchy possesses infinitely commuting symmetries and conservation laws. A conclusion will be given in the last section.


## 2. A Spectral Problem and Its Soliton Hierarchy

To obtain a soliton hierarchy, we introduce a new spectral problem(10)ϕx=Uu,λϕ,u=pq,ϕ=ϕ1ϕ2ϕ3,where the spectral matrix U is defined by(11)U=λ2qe1+λe2+λpe3=0-λp-λ2qλp0-λλ2qλ0∈⁡so~3,R.Once a matrix spectral problem is chosen, we apply the generating procedure [1, 3] to work out a soliton hierarchy associated with the spectral problem. First, we solve the stationary zero curvature equation(12)Wx=U,W.If we assume W to be (13)W=ae1+be2+ce3=0-c-ac0-bab0,then (12) gives(14)ax=λc-pb,bx=-λ2qc+λpa,cx=-λa+λ2qb.Further, let a, b, and c possess the Laurent expansions: (15)a=∑i≥0aiλ-2i,b=∑i≥0biλ-2i-1,c=∑i≥0ciλ-2i-1,and then, the system (14) leads to (16)aix=ci-pbi,bix=pai+1-qci+1,cix=qbi+1-ai+1,i≥0.Using the above recursion relations (16) and taking the initial data(17)a0=1,b0=1q,c0=pq,the sequences of {ai,bi,ci∣i≥1} are determined uniquely provided that we take zero constants of integration. The first two sets are (18)a1=-1+p22q2,b1=2qpx-2pqx-p2-12q3,c1=2qx-p-p32q3;a2=3+6p2+3p4-8qpx8q4,b2=18q531+p21+p2-4qpx+4pqx-24qx2+8qqxx,c2=18q56p3+3p5-12p2qx+122qpx-1qx-8q2pxx+p3-24qx2+8qqxx.From (16), we have(19)λλ2m+1W+x-U,λλ2m+1W+=λpam+1-qcm+1e2+λqbm+1-am+1e3,m≥0,where p+ denotes the polynomial part of p. However, a direct calculation gives(20)Ut=λpte3+λ2qte1.So in order to work out a soliton hierarchy, we should introduce modification terms Δm, m≥0. Suppose(21)Δm=λ2δ1me1+λδ2me2+λδ3me3,m≥0;we have (22)Δmx-U,Δm=λ2δ3m-λ2pδ2m+λ2δ1mxe1+λ3pδ1m-λ3qδ3m+λδ2mxe2+λ3qδ2m-λ3δ1m+λδ3mxe3.Based on (19)–(22), δ1m, δ2m, and δ3m should satisfy (23)pδ1m-qδ3m=0,qδ2m-δ1m=0,δ2mx+pam+1-qcm+1=0,m≥0.The third equation in (23) gives δ2mx=qcm+1-pam+1=-bmx, so δ2m=-bm, δ1m=-qbm, and δ3m=-pbm; that is,(24)Δm=λ2δ1me1+λδ2me2+λδ3me3=0λpbmλ2qbm-λpbm0λbm-λ2qbm-λbm0,m≥0.As usual, we define(25)Vm=λλ2m+1W++Δm,m≥0,and then the corresponding zero curvature equations(26)Utm-Vxm+U,Vm=0,m≥0,give rise to a soliton hierarchy(27)utm=Km=pqtm=cmx-pbmx-qbmx,m≥0.Remark 1.
The Lax pair for the hierarchy (27) is given by (2) and (3), where U and V are determined by (11) and (25). This implies that the hierarchy (27) is integrable in the Lax sense.The first nonlinear system in this soliton hierarchy (27) is as follows:(28) u t 1 = p q t 1 = 1 q 4 1 + p 2 q q x x - 3 q x 2 - q 2 p x 2 + p p x x + 4 p q p x q x  1 q 3 2 q p x - 1 q x + p q p x + q x x - p 2 q x - q 2 p x x - 2 p q x 2 .


## 3. Bi-Hamiltonian Structures and Liouville Integrability

In this subsection, we will show that the soliton hierarchy (27) is Liouville integrable [16–18]. First, let us establish bi-Hamiltonian structures for the hierarchy (27) by using the trace identity [11, 19] or the variational identity [20, 21]: (29)δδu∫tr∂U∂λWdx=λ-γ∂∂λλγtr∂U∂uW,γ=-λ2ddλln⁡trW2.It is easy to find that(30)∂U∂λ=0-p-2λqp0-1λq10,∂U∂p=0-λ0λ00000,∂U∂q=00-λ2000λ200.Thus, we have(31)trW∂U∂λ=-4λqa-2pc-2b,trW∂U∂p=-2λc,trW∂U∂q=-2λ2a.Plugging these quantities into the trace identity (29) gives (32)δδu∫-4λqa-2pc-2bdx=λ-γ∂∂λλγ-2λc-2λ2a.Balancing the coefficients of λ-2m-1 in the above equation leads to(33)δδu∫2qam+1+pcm+bmdx=γ-2mcmam+1,m≥0.The identity with m=1 yields γ=0 and thus we have(34)δδuHm=cmam+1,m≥0,with the Hamiltonian functions being defined by(35)H0=∫1+p22qdx,Hm=-∫2qam+1+pcm+bm2mdx,m≥1.From (16), we have(36)bm=-∂-1qcm+1+∂-1pam+1=-∂-1q∂am+1+pbm+1+∂-1pam+1=-∂-1q∂am+1-∂-1pcmx+am+1+∂-1pam+1,m≥0.Thus the hierarchy (27) can be written as (37)utm=Km=cmx-pbmx-qbmx=cmx+∂p∂-1q∂am+1+∂p∂-1p∂cm∂q∂-1q∂am+1+∂q∂-1p∂cm=Jcmam+1,m≥0,where(38)J=∂+∂p∂-1p∂∂p∂-1q∂∂q∂-1p∂∂q∂-1q∂.It can be verified that J is skew symmetry and satisfies the Jacobi identity, so it is a Hamiltonian operator. It now follows that the soliton hierarchy (27) has the Hamiltonian structures(39)utm=Km=JδHmδu,m≥0.From the recursion relations (16), we can have(40)cmam+1=Ψcm-1am,m≥0,where(41) Ψ = p q ∂ ∂ + p q  - ∂ - 1 1 q ∂ 1 q ∂ - ∂ - 1 p q ∂ p q ∂ - ∂ - 1 1 q ∂ 1 q - ∂ - 1 p q ∂ 2 - ∂ - 1 p q ∂ p q .It is easy to verify that JΨ=Ψ∗J. Actually, we can show that the hierarchy (27) is bi-Hamiltonian:(42)utm=Km=JδHmδu=MδHm-1δu,m≥1,where the second Hamiltonian operator is(43)M=JΨ=0∂2-∂2-∂.Here J and M constitute a Hamiltonian pair. Particularly, (44)Hk,HlJ=∫δHkδuTJδHlδudx=0,k,l≥0,Hk,HlM=∫δHkδuTMδHlδudx=0,k,l≥0,Kk,Kl=Kk′uKl-Kl′uKk=0,k,l≥0.These commuting relations are also consequences of the Virasoro algebra of Lax operators [22, 23]. To sum up the above discussion, we obtain the following proposition and theorem.Proposition 2.
The soliton hierarchy (27) has infinitely many common commuting symmetries {JδHm/δu}m=1∞ and infinitely many conserved functions {Hm}m=0∞.Theorem 3.
The soliton equations in the soliton hierarchy (27) are all integrable in Liouville sense.


## 4. Concluding Remarks

Based on the special orthogonal Lie algebraso~(3,R), the AKNS spectral matrix, the KN spectral matrix, and the WKI spectral matrix were presented in [11, 16, 24], respectively. Those spectral matrices are (45)Uu,λ=λe1+qe2+pe3,Uu,λ=λ2e1+λqe2+λpe3,Uu,λ=λe1+λqe2+λpe3,Uu,λ=λ+qe1+pe2+e3,where u=(p,q)T includes two dependent variables.In our paper, the new spectral problem defined by (10) with (11), associated with so~(3,R), has been proposed and its corresponding soliton hierarchy has been worked out. The new soliton hierarchy is bi-Hamiltonian, and so, the resulting equations possess infinitely many commuting symmetries and conserved functionals, which implies that they are Liouville integrable. Furthermore, we will further research the integrable couplings and Darboux transformation of the integrable hierarchy (27).It is particularly interesting to explore other types of spectral problems of even higher degree in a spectral parameter, associated withso(3,R), which generate soliton hierarchies. All those studies should provide insightful thoughts to classify multicomponent integrable systems.



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*Source: 3589704-2016-03-30.xml*