# The Hamiltonian Structure and Algebrogeometric Solution of a 1 + 1-Dimensional Coupled Equations

Authors: Hongfei Pan; Tiecheng Xia
Journal: Journal of Applied Mathematics
 (2013)

---

## Abstract

A 1 + 1-dimensional coupled soliton equations are decomposed into two systems of ordinary differential equations. The Abel-Jacobi coordinates are introduced to straighten the flows, from which the algebrogeometric solutions of the coupled 1 + 1-dimensional equations are obtained in terms of the Riemann theta functions.


---

## Body

## 1. Introduction

The study of explicit solutions for soliton equations is very important in modern mathematics, physics, and other sciences. There are several systematic approaches to obtain explicit solutions of the soliton equations, such as the inverse scattering transformation, the algebrogeometric method, and the Hirota bilinear method (see, e.g., [1–4] and references therein). Algebrogeometric (or quasiperiodic) solutions are very important explicit solutions for soliton equations, recently, based on the nonlinearization technique of Lax pairs and direct method proposed by Cao [5]. This new scheme is further shown to be a very powerful tool, through which algebrogeometric solutions of (1+1)-dimensional and (2+1)-dimensional continuous and discrete soliton equations can be obtained; see Cao et al. [6, 7], Geng et al. [8, 9], and Dai and Fan [10].In this paper, we will construct the Hamiltonian structure and search for the algebrogeometric solution of the following coupled1+1-dimensional soliton equations:
(1)qt=rxx-3q2rx+qxr2+2qrrx,rt=qxx+3r2rx-rxq2-2qrqx.Our purpose is to construct the Hamiltonian structure and give the algebrogeometric solutions of the coupled1+1-dimensional soliton equations based on its obtained Lax pairs. The paper is organized as follows. In Section 2, we use Lenard operator pairs to derive another form of the coupled 1+1-dimensional soliton equations. In Section 3, based on the trace identity [11, 12], we construct the Hamiltonian structure of the coupled 1+1-dimensional soliton equations. In Section 4, based on the Lax pairs of the coupled 1+1-dimensional soliton equations, variable separation technique is used to translate the solution of the coupled 1+1-dimensional soliton equations to solve ordinary differential equations. In Section 5, a hyperelliptic Riemann surface of genus N and Abel-Jacobi coordinates are defined to straighten the associated flows. Jacobi's inverse problem is discussed, from which the algebrogeometric solutions of the coupled 1+1-dimensional soliton equations are constructed in terms of the Riemann theta functions.


## 2. The Hierarchy and Lax Pairs of the Coupled 1 + 1-Dimensional Soliton Equations

In this section, we introduce the Lenard gradient sequence{Sj}j=0,1,2… to derive the hierarchy and its stationary hierarchy associated with (1) by the recursion relation:
(2)KSj-1=JSj,j=1,2,3,….Sj|(u,v)=0=0,S0=(q-r,q+r,1)T,
where Sj=(Sj(1),Sj(2),Sj(3)) and operators (∂=∂/∂x)
(2)K=(∂000∂0-(q+r)q-r∂),J=(-202(q-r)02-2(q+r)-(q+r)q-r∂).
A direct calculation gives from the recursion relation (2) that(4)S1=(-12(q+r)(q-r)2-12(qx-rx)-12(q+r)2(q-r)+12(qx+rx)-12(q+r)(q-r)),S2=(38(q+r)2(q-r)3+12r(q-r)(qx-rx)+12(q+r)(q-r)(qx-rx)+14(qxx-rxx)+14(qx+rx)(q-r)2-12(q-r)∂-1(qx-rx)rx38(q+r)3(q-r)2+12r(q+r)(qx-rx)-12(q+r)(q-r)(qx+rx)-14(qxx-rxx)-14(qx-rx)(q+r)2-12(q+r)∂-1(qx-rx)rx38(q+r)2(q-r)2+12r(qx-rx)-12∂-1(qx-rx)rx).Consider the spectral problem
(5)ψx=Uψ,U=(λq+rλ(q-r)-λ),
and the auxiliary problem
(6)ψtm=V(m)ψ,V(m)=(V11(m)V12(m)V21(m)-V11(m)),
where
(7)V11(m)=∑j=0m‍Sj(3)λm+1-j,V12(m)=∑j=0m‍Sj(2)λm-j,V21(m)=∑j=0m‍Sj(1)λm+1-j.Then the compatibility condition of (5) and (6) is Utm-Vx(m)+[U,V(m)]=0, which is equivalent to the hierarchy of nonlinear evolution equations
(8)qtm=12(Smx(2)+Smx(1)),rtm=12(Smx(2)-Smx(1)).In brief,(9)(qtm,rtm)T=Xm,m≥0,(10)Xm=(∂∂∂-∂)(Sm(2)Sm(1)).The first two nontrivial equations are(11)qt0=qx,rt0=rx,qt1=rxx-3q2rx+qxr2+2qrrx,rt1=qxx+3r2rx-rxq2-2qrqx.The second system is our coupled1+1-dimensional soliton equations (1).Letψ=(ψ1,ψ2)T and ϕ=(ϕ1,ϕ2)T be two basic solutions of the spectral equations (5) and (6). We define a matrix W by
(12)W=12(ϕψT+ψϕT)σ=(fgh-f),σ=(0-110),
in which f,g, and h are three functions. It is easy to calculate by (5) and (6) that
(13)Wx=[U,W],Wtm=[V(m),W],
which implies that ∂xdetW=0,∂tmdetW=0. Equation (13) can be written as
(14)fx=-λ(q-r)g+(q+r)h,gx=-2(q+r)f+2λg,hx=2λ(q-r)f-2λh,(15)ftm=hV12(m)-gV21(m),gtm=2gV11(m)-2fV12(m),htm=2fV21(m)-2hV11(m).We suppose that the functionsf,g, and h are finite-order polynomials in λ:
(16)f=∑j=0N‍fjλN-j+1,g=∑j=0N‍gjλN-j,h=∑j=0N‍hjλN-j.Substituting (16) into (14) yields
(17)KGj-1=JGj(j=1,2,…,N),JG0=0,KGN=0,Gj=(hj,gj,fj)T.It is easy to see that (17) implies
(18)-(q+r)hj+(q-r)gj+fjx=0
and the equation JG0=0 has the general solution
(19)G0=α0S0,
where α0 is constant of integration. Therefore, if we take (19) as a starting point, then Gj can be determined recursively by relation (17). In fact, noticing kerJ={cS0|∀c∈ℝ} and acting with the operator (J-1K)k upon (19), we obtain from (2) and (17) that
(20)Gk=∑j=0k‍αjSk-j,k=0,1,…,N,
where α0,α1,…,αk are integral constants. Substituting (20) into (17) yields a certain stationary evolution equation:
(21)α0X^N+α1X^N-1+⋯+αNX^0=0,
where
(22)X^j=(∂00∂)(Sj(1)Sj(2)).This means that expression (16) is existent.


## 3. Hamiltonian Structure

Let(23)V=V=(V11V12V21-V11),
where
(24)V11=∑j≥0‍Sj(3)λ-j+1,V12=∑j≥0‍Sj(2)λ-j,V21=∑j≥0‍Sj(1)λ-j+1.It is easy to calculate(25)tr(V∂U∂λ)=2V11+(q-r)V12=∑j≥1‍(2Sj(3)+(q-r)Sj-1(2))λ-j+1+2S0(3)λ,tr(V∂U∂q)=λV12+V21=∑j≥0‍(Sj(2)+Sj(1))λ-j+1,tr(V∂U∂r)=-λV12+V21=∑j≥0‍(-Sj(2)+Sj(1))λ-j+1.According to the trace identity [11, 12], we have
(26)(δδqδδr)(2V11+(q-r)V12)=(λ-s∂∂λλs)(λV12+V21-λV12+V21).Comparing the coefficients ofλ-j+1, we obtain
(27)(δδqδδr)(2Sj(3)+(q-r)Sj-1(2))=(-j+2+s)(Sj-1(2)+Sj-1(1)-Sj-1(2)+Sj-1(1)),
we set j=1 and then get s=-1 and
(28)(δδqδδr)ℋj=(Sj-1(2)+Sj-1(1)-Sj-1(2)+Sj-1(1))=(11-11)(Sj-1(2)Sj-1(1)),
where ℋj=(2Sj(3)+(q-r)Sj-1(2))/(-j+1).Thus the soliton equation (9) has a Hamiltonian structure:
(29)(qr)tm=12(∂∂∂-∂)(Sm(2)Sm(1))=J~(δδqδδr)ℋm+1,m≥0,
where
(30)J~=12(∂00-∂).In speciality, the Hamiltonian structure of (1) is (m=1):
(31)ℋ2=18(q+r)2(q-r)2-qqx-2rrx+qrx2+∂-1(qx-rx)rx2.


## 4. Ordinary Differential Equations

In this section, (1) will be decomposed into two systems of solvable ordinary differential equations. Without loss of generality, let α0=1. From (2), (17), and (20), we have
(32)f0=1,g0=q+r,h0=q-r,f1=-12(q+r)(q-r)+α1,g1=-12(q+r)2(q-r)+12(qx+rx)+α1(q+r),h1=-12(q+r)(q-r)2-12(qx-rx)+α1(q-r).By using (16), we can write g and h as the following finite products:
(33)g=-(q+r)∏j=1N‍(λ-uj),h=(q-r)∏j=1N‍(λ-vj).Equation (33) implies by comparing the coefficients of λN-1 that
(34)g1=-(q+r)∑j=1N‍uj,h1=-(q+r)∑j=1N‍vj.
Thus from (32) and (34), we obtain
(35)-12(q+r)2(q-r)+12(qx+rx)+α1(q+r)=-(q+r)∑j=1N‍uj,-12(q+r)(q-r)2-12(qx-rx)+α1(q-r)=-(q-r)∑j=1N‍vj.Let us consider the functiondetW which is a (2N+2)th-order polynomial in λ with constant coefficients of the x-flow and tm-flow:
(36)-detW=f2+gh=∏j=12N+2‍(λ-λj)=R(λ).
Substituting (16) into (36) and comparing the coefficients of λ2N+1 yield
(37)2f0f1+g0h0=-∑j=12N+2‍λj,
which together with (32) gives
(38)α1=-12∑j=12N+2‍λj.From (36) we see that
(39)f|λ=uk=R(uk),f|λ=vk=R(vk).Again by using (14) and (33), we obtain
(40)gx|λ=uk=-(q+r)ukx∏j=1,j≠kN‍(uk-uj)=-2(q+r)f|λ=uk,hx|λ=vk=-(q-r)vkx∏j=1,j≠kN‍(vk-vj)=2vk(q-r)f|λ=vk,
which together with (39) gives
(41)ukx=2R(uk)∏j=1,j≠kN(uk-uj),1≤k≤N,vk,x=-2R(vk)∏j=1,j≠kN(vk-vj),1≤k≤N.In a way similar to the above expression, by using (6) (m=1,t1=t), (15), and (39), we arrive at the evolution of {uk} and {vk} along the tm-flow:
(42)uk,t=2f|λ=ukV12(1)|λ=uk(q+r)∏j=1,j≠kN(uk-uj)=2R(uk)[uk-(1/2)(q+r)(q-r)+(1/2)∂ln(q-r)]∏j=1,j≠kN(uk-uj),(43)vk,t=-2f|λ=vkV21(1)|λ=vk(q-r)∏j=1,j≠kN(vk-vj)=(-2R(vk)[vk2-12(q+r)(q-r)vk+12∂ln(q-r)vk])(∏j=1,j≠kN‍(vk-vj))-1.Therefore, if the(2N+2) distinct parameters λ1,λ2,…,λ2N+2 are given and let uk(x,t) and vk(x,t) be distinct solutions of ordinary differential equations (41), (42), and (43), then (q,r) determined by (35) is a solution of the coupled 1+1-dimensional equations (1).


## 5. Algebrogeometric Solutions

In this section, we will give the algebrogeometric solutions of the coupled1+1-dimensional equation (1). To this end, we first introduce the Riemann surface Γ of the hyperelliptic curve
(44)Γ:ζ2=R(λ),R(λ)=∏j=12N+2‍(λ-λj),
with genus N on Γ. On Γ there are two infinite points ∞1 and ∞2, which are not branch points of Γ. We equip Γ with a canonical basis of cycles: a1,a2,…,aN;b1,b2,…,bN which are independent and have intersection numbers as follows:
(45)ai∘aj=0,bi∘bj=0,ai∘bj=δij,i,j=1,2,…,N.We will choose the following set as our basis:(46)ω~l=λl-1dλR(λ),l=1,2,…,N,
which are linearly independent of each other on Γ, and let
(47)Aij=∫aj‍ω~i,Bij=∫bj‍ω~i.It is possible to show that the matricesA=(Aij) and B=(Bij) are N×N invertible matrices [13, 14]. Now we define the matrices C and τ by C=(Cij)=A-1,τ=(τij)=A-1B. Then the matrix τ can be shown to be symmetric (τij=τji) and it has a positive-definite imaginary part (Im τ>0). If we normalize ω~j into the new basis ωj(48)ωj=∑l=1N‍Cjlω~l,l=1,2,…,N,
then we have
(49)∫ajωj=∑l=1N‍Cjl∫aj‍ω~l=∑l=1N‍CjlAli=δji,∫bjωi=∑l=1N‍Cjl∫bj‍ω¯l=∑l=1N‍CjlBli=τji.Now we introduce the Abel-Jacobi coordinates as follows:(50)ρj(1)(x,t)=∑k=1N‍∫p0p(uk(x,t))‍ωj=∑k=1N‍∑l=1N‍∫λ(p0)uk‍Cjlλl-1dλR(λ),(51)ρj(2)(x,t)=∑k=1N‍∫p0p(vk(x,t))‍ωj=∑k=1N‍∑l=1N‍∫λ(p0)vk(x,t)‍Cjlλl-1dλR(λ),
where p(uk(x,t))=(uk,R(uk)),p(vk(x,t))=(vk,R(vk)), and λ(p0) is the local coordinate of p0. From (42) and (50), we get
(52)∂xρj(1)=∑k=1N‍∑l=1N‍Cjlukl-1ukxR(uk)=∑k=1N‍∑l=1N‍Cjlukl-1∏j=1,j≠kN(uk-uj),
which implies
(53)∂xρj(1)=2CjN=Ωj(1),j=1,2,…,N.With the help of the following equality(54)∑k=1N‍ukl-1∏j=1,j≠kN(uk-uj)=δlN,l=1,2,…,N,
in a similar way, we obtain from (50), (51), (41), (42), and (43) that
(55)∂tρj(1)=2∑l=1N‍Cjl(∑i=1N‍ui)-CjN(q+r)(q-r)+CjN∂ln(q-r)=Ωj(2),∂xρj(2)=-Ωj(1),j=1,2,…,N,∂tρj(2)=-Ωj(2),j=1,2,…,N.On the basis of these results, we obtain the following:(56)ρj(1)(x,t)=Ωj(1)x+Ωj(2)t+γj(1),ρj(2)(x,t)=-Ωj(1)x-Ωj(2)t+γj(2),
where γj(i)(i=1,2) are constants and
(57)γj(1)=∑k=1N‍∫p0p(u~k(0,0))‍ωj,γj(2)=∑k=1N‍∫p0p(v~k(0,0))‍ωj,ρ(1)=(ρ1(1),ρ2(1),…,ρN(1))T,ρ(2)=(ρ1(2),ρ2(2),…,ρN(2))T,Ω(m)=(Ω1(m),Ω2(m),…,ΩN(m))T,γ(m)=(γ1(m),γ2(m),…,γN(m))T,m=1,2.Now we introduce the Abel map𝒜(p):
(58)𝒜(p)=∫p0p‍ω,ω=(ω1,ω2,…,ωN)T,𝒜(∑k‍pk)=∑‍nk𝒜(pk)
and Abel-Jacobi coordinates:
(59)ρ(1)=𝒜(∑k=1N‍p(uk))=∑k=1N‍∫p0p(uk)‍ω,ρ(2)=𝒜(∑k=1N‍p(vk))=∑k=1N‍∫p0p(vk)‍ω.
According to the Riemann theorem [13, 14], there exists a Riemann constant vector M∈ℂN such that the function
(60)F(m)(λ)=θ(𝒜(p(λ))-ρ(m)-M(m)),m=1,2,
has exactly N zeros at u1,u2,…,uN for m=1 or v1,v2,…,vN for m=2. To make the function single valued, the surface Γ is cut along all ak,bk to form a simple connected region, whose boundary is denoted by γ. By [13, 14], the integrals
(61)I(Γ)=12πi∫γ‍λdlnF(m),m=1,2,
are constants independent of ρ(1) and ρ(2) with
(62)I=I(Γ)=∑j=1N‍∫aj‍λωj.By the residue theorem, we have(63)∑j=1N‍uj=I-∑s=12‍Resλ=∞sλdlnF(1)(λ),∑j=1N‍vj=I-∑s=12‍Resλ=∞sλdlnF(2)(λ).Here we need only to compute the residues in (63). In a way similar to calculations in [10], we arrive at
(64)Resλ=∞sλdlnF(m)(λ)=(-1)s+m∂lnθs(m),m=1,2;s=1,2,
where
(65)θs(1)=θ(Ω(1)x+Ω(2)t+ξs),θs(2)=θ(-Ω(1)x-Ω(2)t+ηs),
and ξs and ηs are constants. Thus from (63) and (64), we arrive at
(66)∑j=1N‍uj=I-∂lnθ1(1)θ2(1),∑j=1N‍vj=I-∂lnθ2(2)θ1(2).Substituting (66) into (35), we get an algebrogeometric solution for the coupled 1+1-dimensional soliton equations (1):
(67)q=A(t)2exp(-∂-1∑j=12N+2‍λj+2∂-1I-2lnθ2(2)θ1(2)-∂-1exp(-2lnθ2(2)θ2(1)θ1(2)θ1(1)))+A(t)2exp(∂-1∑j=12N+2‍λj+2∂-1I-2lnθ1(2)θ2(1)+∂-1exp(-2lnθ2(2)θ2(1)θ1(2)θ1(1))),(68)r=A(t)2exp(∂-1∑j=12N+2‍λj+2∂-1I-2lnθ1(2)θ2(1)+∂-1exp(-2lnθ2(2)θ2(1)θ1(2)θ1(1)))-A(t)2exp(-∂-1∑j=12N+2‍λj+2∂-1I-2lnθ2(2)θ1(2)-∂-1exp(-2lnθ2(2)θ2(1)θ1(2)θ1(1))),
where A(t) is arbitrary complex functions about variable t.



---

*Source: 782436-2013-09-26.xml*