# Journal of Nonlinear Mathematical Physics

Volume 25, Issue 1, February 2018, Pages 54 - 65

# Finite genus solutions for Geng hierarchy

Authors
Zhu Li
School of Mathematics and Statistics, Xinyang Normal University, 237 Nanhu Road, Xinyang, Henan 464000, China,lizhu2020@126.com
Received 25 October 2016, Accepted 18 August 2017, Available Online 6 January 2021.
DOI
10.1080/14029251.2018.1440742How to use a DOI?
Keywords
Hyperelliptic curve; meromorphic function; finite genus solutions
Abstract

The Geng hierarchy is derived with the aid of Lenard recursion sequences. Based on the Lax matrix, a hyperelliptic curve 𝒦n + 1 of arithmetic genus n+1 is introduced, from which meromorphic function ϕ is defined. The finite genus solutions for Geng hierarchy are achieved according to asymptotic properties of ϕ and the algebro-geometric characters of 𝒦n + 1.

Open Access

## 1. Introduction

The soliton equations describe various nonlinear phenomena in natural and applied sciences such as fluid dynamics, plasma physics, optical fibers and other sciences. It is of great importance to solve nonlinear soliton equations from both theoretical and practical points of view. Due to the nonlinearity of soliton equations, it is a difficult job for us to determine whatever exact solutions to soliton equations, but with the development of soliton theory several systematic methods has been developed to obtain explicit solutions of soliton equations, such as the inverse scattering transformation [1], the Hirota bilinear transformation [2], the Bäcklund and the Darboux transformation [3,4], the algebro-geometric method [5], the nonlinearization approach of Lax pairs [6], the homogeneous balance method [7], etc [812].

The nonlinear diffusion equation [1315]

ut=(uxu2)x, (1.1)
which has a lot of applications in plasma physics, laser physics, semiconductor physics, astrophysics, population dynamics in biology and helium combustion process in chemistry, can be obtained in the following equations in case of v = 1[16]
{ut=(1u2)x-(v2u2)x-(vu)xx,vt=12(1u2)xx-12(v2u2)xx. (1.2)

In this paper we will concentrate primarily on constructing the finite genus solutions of the entire Geng hierarchy related to (1.2) based on the approaches in Refs. [1732]. The finite genus solutions are associated to nonłlinear flows in the Jacobian of a hyperelliptic curve. This phenomenon is connected to the existence of integrable hierarchies with nonłlinear dependence on the spectral parameter. Such problem was first considered in Refs. [24] and [25]. The algebraic geometric approach was proposed in [26]. Key examples are the Camassa-Holm [27] and Harry dym equations [28] whose algebraic geometric solutions produce nonłlinear flows in the generalized Jacobian of hyperelliptic curves in Refs. [29] and [30]. After separation of variables, the appearance of nonłlinear flows in the (generalized) Jacobians of algebraic curves, also appears in ODEs and was first considered in Refs [31] and [32].

The outline of this paper is as follows. In Section 2, we obtain the coupled diffusion equation hierarchy based on Lenard recursion sequences. In Section 3, with the aid of Lax matrix we shall introduce hyperelliptic curve 𝒦n + 1 of arithmetic genus n + 1. In Section 4, we define the meromorphic function ϕ and investigate the asymptotic properties of ϕ. Moreover, we construct the finite genus solutions of the whole hierarchy by use of the Riemann theta functions according to the asymptotic properties of ϕ and the algebro-geometric characters of 𝒦n + 1.

## 2. Hierarchy of nonlinear evolution equations

In this section, we shall derive the Geng hierarchy associated with the 2 × 2 spectral problem [16]

ϕx=Uϕ,ϕ=(ϕ1ϕ2),U=(λuv-1λ(v+1)-λu), (2.1)
where u and v are two potentials, and λ is a constant spectral parameter. To this end, we first introduce the Lenard recursion sequences
KLj=JLj+1,Lj=(aj,bj)T,j0 (2.2)
with three starting points
L0=(1-v22u2,vu)T,L¯0=(1,0)T,L˜0=(0,1)T, (2.3)
where K and J are two operators defined by
K=(2-220),J=2(u-1uu-1vv-1uv-1v-),-1=-1=1. (2.4)

It is easy to see that KerJ={α0L0+α¯0L¯0+α˜0L˜0α0,α¯0,α˜0𝕉} . Hence Lj are uniquely determined by the recursion relation (2.2) up to a term const. L0 + const. L¯0+ const. L˜0 , which is always assumed to be zero. Assume that the eigenfunction φ satisfies an auxiliary problem

ϕtm=V(m)ϕ,V(m)=(V11(m)V12(m)V21(m)-V11(m)), (2.5)
where V11(m),V12(m),V21(m) are defined as follows;
V11(m)=j=0m(-bj-2u-1uajλ-2u-1vbjλ)λm+1-j,V12(m)=j=0m(aj+2-1uajλ-2v-1uajλ+2-1vbjλ-2v-1vbjλ)λm-j,V21(m)=j=0m(aj-2-1uajλ-2v-1uajλ-2-1vbjλ-2v-1vbjλ)λm+1-j. (2.6)

Then the compatibility condition of (2.1) and (2.5) yields the zero curvature equation, Utm-Vx(m)+[U,V(m)]=0 , which is equivalent to a class of nonlinear evolution equations

utm=2amx-bm,xx,vtm=am,xx. (2.7)

The first two nontrivial flows in (2.7) are (1.2) and

ut=1u(1-v2+vux-uvxu3)x-vu(2ux-v2ux+uvvx+v-v3u3)x-12(2ux-v2ux+uvvx+v-v3u3)xx,vt=12(1u(1-v2+vux-uvxu3)x-vu(2ux-v2ux+uvvx+v-v3u3)x)x. (2.8)

## 3. Hyperelliptic curve

Let χ = (χ1, χ2)T and ψ = (ψ1, ψ2)T be two basic solutions of (2.1) and (2.5). We introduce a Lax matrix

W=12(χψT+ψχT)(0-110)=(GFH-G) (3.1)
which satisfies the Lax equations
Wx=[U,W],Wtm=[V(m),W]. (3.2)

Therefore, detW is a constant independent of x and tm. Equation (3.2) can be written as

Gx=(v-1)H-λ(v+1)F,Fx=2λuF-2(v-1)G,Hx=2λ(v+1)G-2λuH, (3.3)
and
Gtm=V12(m)H-V21(m)F,Ftm=2(V11(m)F-V12(m)G),Htm=2(V21(m)G-V11(m)H). (3.4)

Suppose functions F, G and H are finite-order polynomials in λ

G=j=0n+1Gjλn+2-j,F=j=0n+1Fjλn+1-j,H=j=0n+1Hjλn+2-j, (3.5)
where
G0=-2u-1uc0-2u-1vd0,F0=2(1-v)-1uc0+2(1-v)-1vd0,H0=-2(1+v)-1uc0-2(1+v)-1vd0,Fn+1=Hn+1=cn,x,Gj=-dj-1-2u-1ucj-2u-1vdj,1jn,Gn+1=-dn,x,Fj=cj-1+2(1-v)-1ucj+2(1-v)-1vdj,1jn,Hj=cj-1-2(1+v)-1ucj-2(1+v)-1vdj,1jn. (3.6)

Substituting (3.5) and (3.6) into (3.3) yields

KEj=JEj+1,JE0=0, (3.7)
KEn=0, (3.8)
where Ej = (cj, dj)T, 0 ≤ jn − 1. It is easy to see that the equation JE0 = 0 has a special general solution
E0=L0. (3.9)

By induction, we obtain from recursive relations (3.7) and (2.2) that

Ek=j=0kαjLk-j,0kn, (3.10)
which are special solutions of (3.7), where α1, α2,…,αk are constants of integration and α0 = 1. Moreover, from (3.8) we can get
cn=β0x+β1,dn=β0x2+β2x+β3, (3.11)
where β0, β1, β2, β3 are constants of integration.

Since detW is a (2n + 4) th-order polynomial in λ, whose coefficients are constants independent of x and tm, we have

-detW=G2+FH=4λj=12n+3(λ-λj)=4R(λ), (3.12)
one is naturally led to introduce the hyperelliptic curve 𝒦n + 1 of arithmetic genus n + 1 defined by
𝒦n+1:y2-R(λ)=0. (3.13)

The curve 𝒦n + 1 can be compactified by joining two points at infinity, P∞ ±, where P∞ +P∞ −. For notational simplicity the compactification of the curve 𝒦n + 1 is also denoted by 𝒦n + 1. Here we assume that the zeros λj of R(λ) in (3.12) are mutually distinct. Then the hyperelliptic curve 𝒦n + 1 becomes nonsingular and irreducible.

We write F and H as finite products which take the form

F=2(1-v)uj=1n+1(λ-μj),H=-2(1+v)uλj=1n+1(λ-vj), (3.14)
where {μj}j=1n+1 and {vj}j=1n+1 are called elliptic variables. According to the definition of 𝒦n + 1, we can lift the roots μj and vj to 𝒦n + 1 by introducing
μ^j(x,tm)=(μj(x,tm),-12G(μj(x,tm),x,tm)),j=1,,n+1, (3.15)
v^j(x,tm)=(vj(x,tm),12G(vj(x,tm),x,tm)),j=1,,n+1, (3.16)
where (x, tm) ∈ ℝ2.

From the following lemma, we can explicitly represent αl (0 ≤ ln) by the constants λ1,…,λ2 n + 3.

### Lemma 3.1.

αl=cl(Λ_),l=0,,n, (3.17)
where
Λ_=(λ1,,λ2n+3),c0(Λ_)=1,c1(Λ_)=-12j=12n+3λj,,cl(Λ_)=-j1,,j2n+3=0j1++j2n+3=ll(2j1)!(2j2n+3)!λ1j1λ2n+3j2n+322l(j1!)2(j2n+3!)2(2j1-1)(2j2n+3-1). (3.18)

Proof. Assume that

F^j=Fj|α1==αj=0,H^j=Hj|α1==αj=0,G^j=Gjα1==αj=0. (3.19)

It will be convenient to introduce the notion of a degree, deg(.), to effectively distinguish between homogeneous and nonhomogeneous quantities. Define

deg(u)=-1,deg(v)=0,deg(x)=1, (3.20)
thus from (3.7) it can be implied that
deg(F^k)=deg(H^k)=2k+1,deg(G^k)=2k,k0. (3.21)

Temporarily fixed the branch of R(λ)1 / 2 as λn + 2 near infinity, R(λ)−1 / 2 has the following expansion

R(λ)-1/2=λl=0c^l(Λ_)λ-n-2-l, (3.22)
where
Λ=(λ1,,λ2n+3),c^0(Λ_)=1,c^1(Λ_)=12j=12n+3λj,,c^l(Λ_)=j1,,j2n+3=0j1++j2n+3=ll(2j1)!(2j2n+3)!λ1j1λ2n+3j2n+322(j1!)2(j2n+3!)2. (3.23)

Dividing F(λ), H(λ), G(λ) by R(λ)1 / 2 near infinity respectively, we obtain

F(λ)R(λ)1/2=λl=0c^l(Λ_)λ-n-2-ll=0n+1Flλn+1-l=l=0Fˇlλ-l-1,H(λ)R(λ)1/2=λl=0c^l(Λ_)λ-n-2-ll=0n+1Hlλn+2-l=l=0Hˇlλ-l,G(λ)R(λ)1/2=λl=0c^l(Λ_)λ-n-2-ll=0n+1Glλn+2-l=l=0Gˇlλ-l, (3.24)
for some coefficients Fˇl,Fˇl,Gˇl to be determined next. From (3.3) and (3.24), we have
Gˇk,x=(v-1)Hˇk-(v+1)Fˇk,Fˇk,x=2uFˇk+1-2(v-1)Gˇk+1,Hˇk,x=2(v+1)Gˇk+1-2uHˇk+1, (3.25)
for k ∈ ℕ0. The initial values of Fˇ0,Hˇ0 and Gˇ0 have been chosen as Fˇ0=2(1-v)u,Hˇ0=-2(1+v)u,Gˇ0=-2 such that Fˇ0=F^0,Hˇ0=H^0 and Gˇ0=G^0 . Moreover, we can prove inductively that
deg(Fˇk)=deg(Hˇk)=2k+1,deg(Gˇk)=2k,k0. (3.26)

Hence, Fˇl , Hˇl and Gˇl are equal to F^l,H^l and G^l respectively for all l ∈ ℕ0. Thus we proved

F(λ)R(λ)1/2=l=0F^lλ-l-1,H(λ)R(λ)1/2=l=0H^lλ-l,G(λ)R(λ)1/2=l=0G^lλ-l. (3.27)

Considering

R(λ)1/2=λl=0cl(Λ_)λn+2-l, (3.28)
a comparison of the coefficients of λk in the following equation
1=R(λ)1/2×R(λ)-1/2=(l=0cl(Λ_)λn+2-l)(l=0c^l(Λ_)λ-n-2-l) (3.29)
yields
l=0kck-l(Λ_)c^l(Λ_)=δk,0,k0. (3.30)

Therefore, we compute that

m=0kck-m(Λ_)F^m=m=0kck-m(Λ_)l=0mFlc^m-l(Λ_)=l=0kFlp=0k-lck-l-p(Λ_)c^p(Λ_)=Fk, (3.31)
where k = 0, …, n.

## 4. Finite genus solutions

Equip the 𝒦n + 1 with canonical basis cycles: a˜1,,a˜n+1;b˜1,,b˜n+1 , which are independent and have intersection numbers as follows

a˜ja˜k=0,b˜jb˜k=0,a˜jb˜k=δjk. (4.1)

For the present, we will choose our basis as the following set [17]

ω˜l=λl-1dλy(P),1ln+1, (4.2)
which are n + 1 linearly independent homomorphic differentials on 𝒦n + 1. Then the period matrices A and B can be constructed from
Akj=a˜jω˜k,Bkj=b˜jω˜k. (4.3)

It is possible to show that matrices A and B are invertible [33,34]. Now we define the matrices C and τ by C = A−1, τ = A−1B. The matrix τ can be shown to be symmetric (τkj = τjk), and it has positive definite imaginary part (Imτ > 0). If we normalize w˜l into the new basis ωj,

ωj=l=1n+1Cjlω˜l,1jn+1, (4.4)
then we obtain
a˜kωj=l=1n+1Cjla˜kω˜l=δjk,b˜kωj=τjk. (4.5)

Let 𝒯n + 1 be the period lattice 𝒯n+1={z_𝔺n+1n_+m_τ;m_,n_𝕑n+1} . The complex torus 𝒯= ℂn + 1 / 𝒯n + 1 is called the Jacobian variety of 𝒦n + 1. Now we introduce the Abel map 𝒜_(P):Div(𝒦n+1)𝒯

𝒜_(P)=(Q0Pω_)(mod𝒯n+1),𝒜_(nkPk)=nk𝒜_(Pk), (4.6)
where P,Pk𝒦n+1,ω_=(ω1,,ωn+1) .

Let θ(z) denote the Riemann theta function associated with 𝒦n + 1 [3335]:

θ(z_)=N_𝕑n+1exp{2πi<N_,z_>+πi<N_τ,N_>}, (4.7)
where z_=(z1,,zn+1)𝔺n+1,<N_,z_>=k=1n+1Nkzk,<N_τ,N_>=k,j=1n+1τkjNkNj . For brevity, define the function z_:𝒦n+1×σn+1𝒦n+1𝔺n+1 by
z_(P,Q_)=Λ_-𝒜_(P)+QQ_𝒟(Q)𝒜(Q), (4.8)
where P𝒦n + 1, Q = {Q1,⋯,Qn + 1} ∈ σn + 1 𝒦n + 1, σn + 1𝒦n + 1 denotes the (n + 1) th symmetric power of 𝒦n + 1, and Λ_=(Λ1,,Λn+1) is the vector of Riemann constant defined by
Λj=12(1+τjj)-k=1kjn+1a˜kωkQ0Pωj,j=1,,n+1. (4.9)

Without loss of generality, we choose the branch point Q0=(λj0,0) , j0 ∈ {1, …, 2 n + 3} as a convenient base point, and λ(Q0) is its local coordinate.

By virtue of (3.12) and (3.13) we can define the meromorphic function ϕ(P, x, tm) on 𝒦n + 1:

φ(P,x,tm)=2y-GF=H2y+G, (4.10)
where P = (λ, y) ∈ 𝒦n + 1 \ {P∞ ±}.

### Lemma 4.1.

Suppose that u(x, tm), v(x, tm) ∈ C(ℝ2) satisfy the hierarchy (2.7). Let λj \{0}, 1 ≤ j ≤ 2n + 3, and P = (λ, y) 𝒦n + 1 \ {P +, P +, P0}, where, P0 = (0,0). Then

φ(P,x,tm)=ζ0{1+v2u+O(ζ),asPP+,ζ=λ-1,2u1-vζ-1+O(1),asPP-,ζ=λ-1, (4.11)

and

φ(P,x,tm)ζ0ζ+O(ζ2),asPP0,ζ=σλ12,σ=±1. (4.12)

Proof. From (3.12) and (3.13), we have

y=ζ0ζ-n-2(1+2G0G1+F0H08ζ+O(ζ2)),asPP±. (4.13)

From (3.5), we obtain

G=ζ0ζ-n-2(G0+G1ζ+O(ζ2)),asPP±, (4.14)
F=ζ0ζ-n-1(F0+F1ζ+O(ζ2)),asPP±. (4.15)

Then according to the definition of ϕ(P, x, tm) in (4.10), we have

φ(P,x,tm)=2y-GF=ζ02-G0+(2G0G1+F0H04-G1)ζ+O(ζ2)ζ(F0+F1ζ+O(ζ2))=ζ0{1+v2u+O(ζ),asPP+,2u1-vζ-1+O(1),asPP-. (4.16)

To prove (4.12), we introduce the local coordinate ζ=σλ12 near P0. Similarly we have

y=ζ012Fn+1ζ+O(ζ3),asPP0, (4.17)
G=ζ0Gn+1ζ2+O(ζ4),asPP0, (4.18)
F=ζ0Fn+1+O(ζ2),asPP0, (4.19)
then (4.12) is given by virtue of (4.10) and (4.17)(4.19).

The divisor of ϕ(P, x, tm) is given by

(φ(P,x,tm))=𝒟P0,v^1(x,tm),,v^n+1(x,tm)-𝒟P-,μ^1(x,tm),,μ^n+1(x,tm) (4.20)
from the Lemma 4.1 and the definition of ϕ(P, x, tm) in (4.10).

Let ωP0,P-(3)(P) denote the normalized Abelian differentials of the third kind holomorphic on 𝒦n +1 \ {P0, P∞ −} with simple poles at P0 and P∞− with residues ± 1, respectively, which can be expressed as

ωP0,P-(3)(P)=12λdλ+12yj=1n+1(λ-δj)dλ, (4.21)
where γi ∈ ℂ, j = 1, …, n + 1, are constants that are determined by
a˜jωP0,P-(3)(P)=0,j=1,,n+1. (4.22)

The explicit formula (4.21) then implies

ωP0,P-(3)(P)=ζ0{(ζ-1+O(1))dζ,asPP0,ζ=σλ12,O(1)dζ,asPP+,ζ=λ-1,(-ζ-1+O(1))dζ,asPP-,ζ=λ-1. (4.23)

Therefore,

Q0PωP0,P-(3)(P)=ζ0{lnζ+ln(ω0)+O(ζ),asPP0,ln(ω+)+O(ζ),asPP+,-lnζ+ln(ω-)+O(ζ),asPP-, (4.24)
for some constants ω∞ +, ω∞−, ω0 ∈ ℂ.

### Theorem 4.1.

Let P = (λ, y) 𝒦n + 1 \ {P +, P∞−, P0},(x, tm) ∈ M, where M2 is open and connected. Suppose u(x, tm), v(x, tm) ∈ C(M) satisfy the hierarchy of equations (2.7), and assume that λj, 1 ≤ j ≤ 2 n + 3, in (3.12) satisfy λj ∈ ℂ \ {0}, and λj ≠ λk as j ≠ k. Moreover, suppose that 𝒟μ^_(x,tm) or equivalently, 𝒟v_^(x,tm) , is nonspecial for (x, tm) ∈ M. Then

1u=ω+ω0θ(z_(P+,v_^(x,tm)))θ(z_(P0,μ^_(x,tm)))θ(z_(P+,μ^_(x,tm)))θ(zi(P0,v_^(x,tm)))+ω0ω-θ(z_(P-,μ^_(x,tm)))θ(z_(P0,v_^(x,tm)))θ(z_(P-,v_^(x,tm)))θ(z_(P0,μ^_(x,tm))), (4.25)
1-v1+v=(ω0)2ω+ω-θ(z(P+,μ^(x,tm)))θ(z_(P-,μ^(x,tm)))θ2(z_(P0,v_^(x,tm)))θ(z_(P+,v_^(x,tm)))θ(z_(P-,v_^(x,tm)))θ2(z_(P0,μ^(x,tm))). (4.26)

Proof. According to Riemann’s vanishing theorem [17,33], the definition and asymptotic properties of ϕ(P, x, tm), ϕ(P, x, tm) has expression of the following type

φ(P,x,tm)=N(x,tm)θ(zz(P,v_^(x,tm)))θ(z_(P,μ^(x,tm)))exp(Q0PωP0,P-(3)(P)), (4.27)
where N(x, tm) is independent of P𝒦n+1,μ^_(x,tm)={μ^1(x,tm),,μ^n+1(x,tm)},v_^(x,tm)={v^1(x,tm),,v^n+1(x,tm)}σn+1𝒦n+1 . Considering the asymptotic expansions of ϕ(P, x, tm) near P∞ ±and P0, we have
1+v2u=N(x,tm)ω+θ(z_(P+,v_^(x,tm)))θ(z_(P+,μ^_(x,tm))), (4.28)
2u1-v=N(x,tm)ω-θ(z_(P-,v_^(x,tm)))θ(z_(P-,μ_^(x,tm))), (4.29)
1=N(x,tm)ω0θ(z_(P0,v_^(x,tm)))θ(z(P0,μ^_(x,tm))), (4.30)

Then combining (4.28)(4.30) yields (4.25) and (4.26).

## 5. Conclusions

In this paper, Finite genus solutions for Geng hierarchy are constructed, which are very important because they reveal inherent structure mechanism of solutions and describe the quasi-periodic behavior of nonlinear phenomenon or characteristic for the integrability of soliton equations. Moreover, they can be used to find multi-soliton solutions, elliptic function solutions, and others. However, we can’t straighten the flows of the entire soliton hierarchy under the Abel-Jacobi coordinates, we will study it in the future.

## Acknowledgments

This work was supported by the Key Scientific Research Projects of Henan Institution of Higher Education (No.17A110029) and Nanhu Scholars Program for Young Scholars of XYNU.

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[5]ED Belokolos, AI Bobenko, VZ Enolskii, AR Its, and VB Matveev, Algebro-geometric approach to nonlinear integrable equations, Springer, Berlin, 1994.
[23]F Gesztesy, H Holden, J Michor, and G Teschl, Soliton Equations and Their Algebro-Geometric Solutions. Vol. II: (1+1)-Dimensional Discrete Models, Cambridge University Press, Cambridge, 2008.
[27]R Camassa and DD Holm, An integrable shallow water equation with peaked solitons, Acta Math. Sinica, Vol. 6, 1990, pp. 35-41.
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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
25 - 1
Pages
54 - 65
Publication Date
2021/01/06
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2018.1440742How to use a DOI?
Open Access

TY  - JOUR
AU  - Zhu Li
PY  - 2021
DA  - 2021/01/06
TI  - Finite genus solutions for Geng hierarchy
JO  - Journal of Nonlinear Mathematical Physics
SP  - 54
EP  - 65
VL  - 25
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2018.1440742
DO  - 10.1080/14029251.2018.1440742
ID  - Li2021
ER  -