# Algebro-Geometric Solutions of a (2+1)-Dimensional Integrable Equation Associated with the Ablowitz-Kaup-Newell-Segur Soliton Hierarchy

Authors: Xiaohong Chen
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

The (2+1)-dimensional Lax integrable equation is decomposed into solvable ordinary differential equations with the help of known (1+1)-dimensional soliton equations associated with the Ablowitz-Kaup-Newell-Segur soliton hierarchy. Then, based on the finite-order expansion of the Lax matrix, a hyperelliptic Riemann surface and Abel-Jacobi coordinates are introduced to straighten out the associated flows, from which the algebro-geometric solutions of the (2+1)-dimensional integrable equation are proposed by means of the Riemann θ functions.


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## Body

## 1. Introduction

Algebro-geometric solutions are an important class among exact solutions of soliton equations, which can be regarded as explicit solutions of the nonlinear integrable evolution equation and used to approximate more general solutions. Based on the nonlinearization technique of Lax pairs and direct method, many of algebro-geometric solutions of (1+1)-dimensional [1–3], (2+1)-dimensional [4, 5], and differential-difference [5, 6] soliton equations have been obtained, such as the Gerdjikov-Ivanov, modified Kadomtsev-Petviashvili, and Toda lattice equations [7–9]. The existence of infinitely many exact solutions is a reflection of this complete integrability.Many other techniques for finding exact solutions have been also discovered: inverse scattering theory, Darboux transformation, Riemann-Hilbert method, etc. Recently, more exact solutions of soliton equations are found [10–13], and more dynamic behaviors are studied [14–16].Ablowitz-Kaup-Newell-Segur (AKNS) soliton hierarchy is an important class of integrable equations, which can be reduced to Korteweg-de Vries (KdV), modified Korteweg-de Vries (mKdV), sine-Gordon equation hierarchies, etc. The purpose of the paper is to further develop the direct method for constructing algebro-geometric solution of the following (2+1)-dimensional integrable equation [15] which concerns with the AKNS soliton hierarchy [17].
(1)ut=−12uxy+u∂x−1uvy,vt=12vxy−v∂x−1uvy.In fact, system (1) is the Lax integrable equations from the AKNS soliton hierarchy, which has nonisospectral zero curvature representation. Ba¨cklund transformation for a splitting of sl2 and a soliton exact solution for it was obtained [18].The whole paper is organized as follows: in Section2, we use Lenard operator pairs to briefly derive (1+1)-dimensional AKNS soliton hierarchy and give the (2+1)-dimensional integrable equation (1). Then, in Section 3, based on the solutions of the (1+1)-dimensional soliton equations and the elliptic coordinates, the solution of the (2+1)-dimensional integrable equation is reduced to solving ordinary differential equations. In Section 4, a hyperelliptic Riemann surface and Abel-Jacobi coordinates are introduced to straighten the associated flows. The Jacobi’s inversion problem is discussed, from which the algebro-geometric solution of the (2+1)-dimensional integrable equation is obtained in terms of the Riemann theta functions. A short summary is in Section 5.


## 2. The (2+1)-Dimensional Soliton Equation

It is well known that the AKNS soliton hierarchy is isospectral evolution equation hierarchy associated with the spectral problem [17].
(2)ψx=Uψ=−λqrλψ,λt=0,ψ=ψ1,ψ2T.Consider the Lenard gradient sequenceSjj=0∞ by
(3)KSj−1=JSj,Sjq,r=0,0=0,S0=0,0,1T, where Sj=Sj1,Sj2,Sj3 and
(4)K=12∂0−r0−12∂−q−qr∂,J=001001−qr∂.It is easy to see thatSj is uniquely determined by the recursion relation. A direct calculation gives that
(5)S1=−r−q0,S2=−12rx−12qx−12qr,S3=−14rxx+12qr2−14qxx+12q2r14rqx−qrx.The auxiliary spectral of (2) is
(6)ψtn=Vψ=ABC−Aψ=∑j=0nSj3λn−j∑j=0nSj2λn−j∑j=0nSj1λn−j−∑j=0nSj3λn−jψ.The compatibility condition between (2) and (6) is the zero curvature equation:
(7)Utn−Vxn+U,Vn=0,which is equivalent to the hierarchy of soliton equations
(8)Xn=qtnrtn=−2Sn+122Sn+11=0−220Sn+11Sn+12,n=1,2,.⋯The first two nontrivial members in the hierarchy are(9)qt2=12qxx−rq2,rt2=12−rxx+qr2,(10)qt3=14−qxxx+6qrqx,rt3=14rxxx+6rqrx.Lett2=y, t3=t, ux,y,t=qx,y,t, and vx,y,t=rx,y,t in (9) and (10); then, we can obtain the (2+1)-dimensional equation (1) by the use of the following equation:
(11)uvx−uxvx=−2uvy.Therefore, ifq and r are the compatible solutions of (9) and (10), then we can get that u=q and v=r are also the solutions of the (2+1)-dimensional equation (1).


## 3. Variable Separation

In this section, we shall show how the (1+1)-dimensional (9) and (10) are reduced to solvable ordinary differential equations. Assume that (2) and (6) have two basic solutions ψ=ψ1,ψ2T and ϕ=ϕ1,ϕ2T. We define a matrix W of three functions f,g,h by
(12)W=12ϕψT+ψϕTσ=fgh−f,σ=0−110.It is easy to verify by (2) and (6) that
(13)Wx=U,W,Wtn=Vn,W,which imply that the functions detW is a constant independent of x and tm. Equation (13) can be written as
(14)gx=−2gλ−2qf,hx=2hλ+2rf,(15)fx=qh−rg,gt=−2gA−2fB,ht=2hA+2fC,ft=hB−gC.Now, suppose that the functionsf, g, and h are finite-order polynomials in λ:
(16)f=∑j=0N+1fjλN+1−j,g=∑j=0N+1gjλN+1−j,h=∑j=0N+1hjλN+1−j.Substituting (16) into (14) yields
(17)KGj−1=JGj,JG0=0,KGN=0,Gj=gj,hj,fjT.It is easy to see thatJG0=0 has the general solution:
(18)G0=α0S0,where α0 is constant of integration. So, KerJ=cS0∀c. Acting with the operator J−1KK+1 upon (18), we can obtain from (3) and (17) that
(19)Gk=∑j=0kαjSk−j,k=0,1,⋯,where α0, …, αk are integral constants. Substituting (19) into (17) obtains the following stationary evolution equation:
(20)α0KSN+⋯+αNKS0=0.This means that expression (16) is existent.In what follows, we decompose (9) and (10) into systems of integrable ordinary differential equations. Without loss of generality, let α0=1. From (3) and (19), we have
(21)g0=0,g1=−q,g2=12qx−α1q,g3=−14qxx+12q2r+α12qx−α2q,(22)h0=0,h1=−r,h2=−12rx−α1r,h3=−14rxx+12qr2−α12rx−α2r,(23)f0=1,f1=α1,f2=−12qr+α2,(24)f3=14rqx−qrx−α12qr+α3,g4=18qxxx−34qrqx−α1212qxx−q2r+α22qx−α3q,(25)h4=−18rxxx+34qrrx+α12−12rxx+r2q−α22rx−α3r,⋯.We can writeg and h as the following finite products:
(26)g=−q∏i=1Nλ−μi,h=−r∏i=1Nλ−νi.Comparing the coefficients ofλN−1, λN−2, and λN−3, we get
(27)g2=q∑i=1Nμi,h2=r∑i=1Nνi,g3=−q∑i<jμiμj,h3=−r∑i<jνiνj,g4=q∑i<j<kμiμjμk,h4=r∑i<j<kνiνjνk,which together with (23), we obtain
(28)12∂xlnq=∑i=1Nμi+α1,12∂xlnr=−∑i=1Nνi−α1,(29)12∂ylnq=∑i<jμiμj+α1∑i=1Nμi+α12−α2,12∂ylnr=−∑i<jνiνj−α1∑i=1Nνi−α12+α2,(30)12∂tlnq=−∑i<j<kμiμjμk−α1∑i<jμiμj−α12−α2∑i=1Nμi−α13+2α1α2−α3,(31)12∂tlnr=∑i<j<kνiνjνk+α1∑i<jνiνj+α12−α2∑i=1Nνi+α13−2α1α2+α3.Let us consider the functiondetW, which is a 2N+2-order polynomial in λ with constant coefficients of the x flow and tn flow:
(32)−detW=f2+gh=∏j=12N+2λ−λj≡Rλ.Substituting (16) into (32), comparing the coefficient of λ2N+1, λ2N, and λ2N−1, and considering (23), we can obtain
(33)α1=−12∑j=12N+2λj,α2=12∑i<jλiλj−18∑j=12N+2λj2,α3=12∑i<j<kλiλjλk+14∑j=12N+2λj∑i<jλiλj−116∑j=12N+2λj3.From (32) we see that
(34)fλ=μk=Rμk,fλ=νk=Rνk.Using (14) and (26), we get
(35)gxλ=μk=−2qfλ=μk=−qμkx∏i=1,i≠kNμk−μi,hxλ=νk=2rfλ=νk=−rνkx∏i=1,i≠kNνk−νi.Together with (34), we get
(36)μkx=2Rμk∏i=1,i≠kNμk−μi,νkx=−2Rνk∏i=1,i≠kNνk−νi.Similarly, using (6) (n=2, n=3), (16), (26), and (34), we get
(37)μkt2=2Rμk∏i=1,i≠kNμk−μi−μk2+∑i=1Nμi+α1μk−∑i<jμiμj−α1∑i=1Nμi−α12+α2,(38)νkt2=−2Rνk∏i=1,i≠kNνk−νi−νk2+∑i=1Nνi+α1νk−∑i<jνiνj−α1∑i=1Nνi−α12+α2,(39)μkt3=2Rμk∏i=1,i≠kNμk−μi−μk3+∑i=1Nμi+α1μk2−∑i<jμiμj+α1∑i=1Nμi+α12−α2μk+∑i<j<kμiμjμk+α1∑i<jμiμj+α12−α2∑i=1Nμi+α13−2α1α2+α3.Therefore, ifλ1,⋯,λ2N+2 are 2N+2 distinct parameters and μk,νkk=1,⋯,N are compatible solutions of differential equations (36), (37), and (39), then q and r determined by (28) are the compatible solution of (9) and (10), so we can get that u and v are also the solution of the (2+1)-dimensional equation (1).


## 4. Algebro-Geometric Solution

We first introduce the hyperelliptic Riemann surface(40)Γ:ξ2=Rλ,Rλ=∏j=12N+2λ−λj,with genus g=N. On Γ, there are two infinite points ∞1 and ∞2, which are not branch points of Γ. Equip Γ with the canonical basis of cycles a1,⋯,aN,b1,⋯,bN, and the holomorphic differentials
(41)ω~l=λl−1dλRλ,l=1,2,⋯,N.Then, the period matricesA and B are defined by
(42)Aij=∫ajω~i,Bij=∫bjω~i.UsingA and B, we can define the matrices C and τ, where
(43)C=Cij=A−1,τ=τij=CB=A−1B.Then, matrixτ can be shown to be symmetric, and it has positive define imaginary part. We normalize ωj~ into the new basis ωj:
(44)ωj=∑l=1NCjlωl~,l=1,2,⋯,N.which satisfy
(45)∫akωi=∑l=1NCjl∫akωl~=∑l=1NCjlAlk=δjk,∫bkωi=∑l=1NCjl∫bkωl~=∑l=1NCjlBlk=τjk.For a fixed pointp0, then we introduce Abel-Jacobi coordinate as follows:
(46)ρm=ρm1,ρm2,⋯,ρmNT,m=1,2,whose components are
(47)ρ1jx,y,t=∑k=1N∫p0μkx,y,tωj=∑k=1N∑l=1N∫p0μkx,y,tCjlλl−1dλRλ,(48)ρ2jx,y,t=∑k=1N∫p0νkx,y,tωj=∑k=1N∑l=1N∫p0νkx,y,tCjlλl−1dλRλ.From (47) and the fist expression of (36), we get
(49)∂xρ1j=∑k=1N∑l=1NCjlμkl−1μkxRμk=∑k=1N∑l=1NCjlμkl−1∏i≠kμk−μj=2CjN=Ω0j,j=1,⋯,N,with the help of the following equality
(50)∑k=1Nμkl−1∏i=i,i≠kNμk−μi=δlN,l=1,⋯,N.In a similar way, we obtain from (36)–(39), (47), and (48) that
(51)∂yρ1j=2−Cj,N−2+α1Cj,N−1−α12−α2Cj,N=Ω1j,∂tρ1j=2−Cj,N−3+α1Cj,N−2−α12−α2Cj,N−1+α13−2α1α2+α3Cj,N=Ω2j,∂xρ2j=−Ω0j,∂yρ2j=−Ω1j,∂tρ2j=−Ω2j.On the basis of these results, we get the following:(52)ρ1jx,y,t=Ω0jx+Ω1jy+Ω2jt+γ0j,ρ2jx,y,t=−Ω0jx−Ω1jy−Ω2jt+γ1j,where
(53)γ0j=∑k=1N∫p0μk0,0,0ωj,γ1j=∑k=1N∫p0νk0,0,0ωj.An Abel map onΓ is defined as
(54)Ap=∫p0pω,ω=ω1,⋯,ωNT,A∑nkpk=∑nkApk.Consider two special divisors∑k=1Npmkm=1,2, and we have
(55)A∑k=1Np1k=∑k=1NAp1k=∑k=1N∫p0μk~ω=ρ1,A∑k=1Np2k=∑k=1NAp2k=∑k=1N∫p0νk~ω=ρ2,where p1k=μk~,ξμk~,p2k=μk~,ξμk~. The Riemann theta function of Γ is defined as
(56)θζ=∑z∈ZNexpπiτz,z+2πiζ,z,ζ∈CN,where ζ=ζ1,⋯,ζNT,ζ,z=∑j=1Nζjzj. According to the Riemann theorem, there exist two constant vector M1,M2∈CN such that
(57)Fm=θAp−ρm−Mm,m=1,2,has exactly zeros at μ1,⋯,μN for m=1 or ν1,⋯,νN for m=2 and m=3. 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. Notice the fact that the integrals
(58)12πi∫γλkdlnFmλ=IkΓ,k≥1,are constants independent of ρ1,ρ2 with I=IΓ=∑j=1N∫ajλkωj. By the residue theorem, we have
(59)IkΓ=∑l=1Nμlk+∑s=12Resλ=∞sλkdlnF1λ,(60)IkΓ=∑l=1Nνlk+∑s=12Resλ=∞sλkdlnF2λ.Here, we only need to compute the residues in (59) for k=1,2,3. In the way similar to calculations in [1, 2, 4], we obtain
(61)Resλ=∞sλdlnFmλ=Resz=0z−1dlnFmz−1=−1s∂xlnθsm,s=1,2;m=1,2.where θs1=θΩ0x+Ω1y+Ω2t+πs,θs2=θ−Ω0x−Ω1y−Ω2t+ηs, and πs,ηs are constants. Thus from, we arrive at
(62)∑j=1Nμj=I1−∂xlnθ21θ11,∑j=1Nνj=I1−∂xlnθ12θ22.Similarly, we obtain(63)∑j=1Nμj2=I2+12∂ylnθ21θ11−12∂x2lnθ11θ21,∑j=1Nνj2=I2+12∂ylnθ22θ12−12∂x2lnθ11θ21,Then, we can get(64)∂xlnq=2I1−∂xlnθ21θ11+2α1=Θ1,∂xlnr=−2I1−∂xlnθ12θ22−2α1=Λ1,∂ylnq=I1−∂xlnθ11θ212−I2+12∂ylnθ21θ11−12∂x2lnlnθ11θ21+2α1I1−∂xlnθ21θ11+2α12+α2=Θ2,∂ylnr=−I1−∂xlnθ12θ222+I2+12∂ylnθ21θ11−12∂x2lnθ11θ21−2α1I1−∂xlnθ12θ22−2α12+α2=Λ2,∂tlnq=−13I1−∂xlnθ21θ113−I2+12∂ylnθ21θ11−12∂x2lnθ11θ21I1−∂xlnθ21θ11+23I3−14∂tlnθ21θ11−14∂y2lnθ11θ21+14∂xlnθ113+14∂xlnθ213+α1I1−∂xlnθ21θ112−I2+12∂ylnθ21θ11−12∂x2lnθ11θ21+2α12−α2I1−∂xlnθ22θ123+2α13−2α1α2+α3=Θ3,∂tlnr=13I1−∂xlnθ12θ223−I2+12∂ylnθ12θ22−12∂x2lnθ11θ21I1−∂xlnθ12θ22+23I3−14∂tlnθ12θ22+14∂y2lnθ12θ22+14∂xlnθ123+14∂xlnθ223−α1I1−∂xlnθ12θ222−I2+12∂ylnθ22θ12−12∂x2lnθ11θ21−2α12−α2I1−∂xlnθ12θ223−2α13−2α1α2+α3=Λ3.With the help of the above equations, we arrive at the algebro-geometric solution of the (2+1)-dimensional equation (1):
(65)u=q=exp∫0,0,0x,y,tΘ1dx+Θ2dy+Θ3dt+c1,v=r=exp∫0,0,0x,y,tΛ1dx+Λ2dy+Λ3dt+c2,where c1 and c2 are constants.


## 5. Summary

The nonisospectral (2+1)-dimensional breaking soliton system is given by the Lenard gradient sequence for a classical (1+1)-dimensional AKNS spectral problem. Then, the (2+1)-dimensional Lax integrable equation associated with the AKNS soliton hierarchy (1) is decomposed into solvable ordinary differential equations with the help of known (1+1)-dimensional soliton equations. With introducing the hyperelliptic Riemann surface and the Abel-Jacobi coordinates, the flow can be straighten out, and the algebro-geometric solutions of the (2+1)-dimensional soliton system (1) are presented by means of the Riemann θ functions.



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