# Finite genus solutions for Geng hierarchy

- 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

*𝒦*_{n + 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*𝒦*_{n + 1}.- Copyright
- © 2018 The Authors. Published by Atlantis Press and Taylor & Francis
- Open Access
- This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

## 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 [8–12].

The nonlinear diffusion equation [13–15]

*v*= 1[16]

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. [17–32]. 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 *𝒦 _{n}*

_{+ 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

*𝒦*

_{n}_{+ 1}.

## 2. Hierarchy of nonlinear evolution equations

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

*u*and

*v*are two potentials, and

*λ*is a constant spectral parameter. To this end, we first introduce the Lenard recursion sequences

*K*and

*J*are two operators defined by

It is easy to see that
*L _{j}* are uniquely determined by the recursion relation (2.2) up to a term const.

*L*

_{0}+ const.

*φ*satisfies an auxiliary problem

Then the compatibility condition of (2.1) and (2.5) yields the zero curvature equation,

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

## 3. Hyperelliptic curve

Let *χ* = (*χ _{1}*,

*χ*

_{2})

*and*

^{T}*ψ*= (

*ψ*

_{1},

*ψ*

_{2})

*be two basic solutions of (2.1) and (2.5). We introduce a Lax matrix*

^{T}Therefore, det*W* is a constant independent of *x* and *t _{m}*. Equation (3.2) can be written as

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

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

*E*= (

_{j}*c*,

_{j}*d*)

_{j}*, 0 ≤*

^{T}*j*≤

*n*− 1. It is easy to see that the equation

*JE*

_{0}= 0 has a special general solution

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

*α*

_{1},

*α*

_{2},…,

*α*are constants of integration and

_{k}*α*

_{0}= 1. Moreover, from (3.8) we can get

*β*

_{0},

*β*

_{1},

*β*

_{2},

*β*

_{3}are constants of integration.

Since det*W* is a (2*n* + 4) th-order polynomial in *λ,* whose coefficients are constants independent of *x* and *t _{m}*, we have

*𝒦*

_{n}_{+ 1}of arithmetic genus

*n*+ 1 defined by

The curve *𝒦 _{n}*

_{+ 1}can be compactified by joining two points at infinity,

*P*

_{∞ ±}, where

*P*

_{∞ +}≠

*P*

_{∞ −}. For notational simplicity the compactification of the curve

*𝒦*

_{n}_{+ 1}is also denoted by

*𝒦*

_{n}_{+ 1}. Here we assume that the zeros

*λ*of

_{j}*R*(

*λ*) in (3.12) are mutually distinct. Then the hyperelliptic curve

*𝒦*

_{n}_{+ 1}becomes nonsingular and irreducible.

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

*𝒦*

_{n}_{+ 1}, we can lift the roots

*μ*and

_{j}*v*to

_{j}*𝒦*

_{n}_{+ 1}by introducing

*x*,

*t*) ∈ ℝ

_{m}^{2}.

From the following lemma, we can explicitly represent *α _{l}* (0 ≤

*l*≤

*n*) by the constants

*λ*

_{1},…,

*λ*

_{2}

_{n}_{+ 3}.

### Lemma 3.1.

*where*

**Proof.** Assume that

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

Temporarily fixed the branch of *R*(*λ*)^{1 / 2} as *λ ^{n}*

^{+ 2}near infinity,

*R*(

*λ*)

^{−1 / 2}has the following expansion

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

*k*∈ ℕ

_{0}. The initial values of

Hence,
*l* ∈ ℕ_{0}. Thus we proved

Considering

*λ*

^{−}

*in the following equation*

^{k}Therefore, we compute that

*k*= 0, …,

*n*.

## 4. Finite genus solutions

Equip the *𝒦 _{n}*

_{+ 1}with canonical basis cycles:

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

*n*+ 1 linearly independent homomorphic differentials on

*𝒦*

_{n}_{+ 1}. Then the period matrices

*A*and

*B*can be constructed from

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*^{−1}*B*. The matrix *τ* can be shown to be symmetric (*τ _{kj}* =

*τ*), and it has positive definite imaginary part (Im

_{jk}*τ*> 0). If we normalize

*ω*,

_{j}Let *𝒯 _{n}*

_{+ 1}be the period lattice

*𝒯*= ℂ

^{n}^{+ 1}/

*𝒯*

_{n}_{+ 1}is called the Jacobian variety of

*𝒦*

_{n}_{+ 1}. Now we introduce the Abel map

Let *θ*(*z*) denote the Riemann theta function associated with *𝒦 _{n}*

_{+ 1}[33–35]:

*P*∈

*𝒦*

_{n}_{+ 1},

*Q*= {

*Q*

_{1},⋯,

*Q*

_{n}_{+ 1}} ∈

*σ*

^{n}^{+ 1}

*𝒦*

_{n}_{+ 1},

*σ*

^{n}^{+ 1𝒦}

_{n}_{+ 1}denotes the (

*n*+ 1) th symmetric power of

*𝒦*

_{n}_{+ 1}, and

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

By virtue of (3.12) and (3.13) we can define the meromorphic function *ϕ*(*P*, *x*, *t _{m}*) on

*𝒦*

_{n}_{+ 1}:

*P*= (

*λ*,

*y*) ∈

*𝒦*

_{n}_{+ 1}\ {

*P*

_{∞ ±}}.

### Lemma 4.1.

*Suppose that u*(*x, t _{m}*),

*v*(

*x, t*)

_{m}*∈ C*(ℝ

^{∞}*)*

^{2}*satisfy the hierarchy (2.7). Let λ*∈

_{j}*ℂ*\{

*0*}, 1

*≤ j ≤*2

*n*+ 3

*, and P*= (

*λ, y*)

*∈*

*𝒦*_{n}_{+}

*\*

_{1}*{P*

_{∞}_{+}

*, P*

_{∞}_{+}

*, P*= (0

_{0}}, where, P_{0}*,*0)

*. Then*

*and*

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

From (3.5), we obtain

Then according to the definition of *ϕ*(*P*, *x*, *t _{m}*) in (4.10), we have

To prove (4.12), we introduce the local coordinate
*P*_{0}. Similarly we have

The divisor of *ϕ*(*P*, *x*, *t _{m}*) is given by

*ϕ*(

*P*,

*x*,

*t*) in (4.10).

_{m}Let
*𝒦 _{n}*

_{+1}\ {

*P*

_{0},

*P*

_{∞ −}} with simple poles at

*P*

_{0}and

*P*

_{∞−}with residues ± 1, respectively, which can be expressed as

*γ*∈ ℂ,

_{i}*j*= 1, …,

*n*+ 1, are constants that are determined by

The explicit formula (4.21) then implies

Therefore,

*ω*

^{∞ +},

*ω*

^{∞−},

*ω*

^{0}∈ ℂ.

### Theorem 4.1.

*Let P* = (*λ, y*) *∈ 𝒦_{n}*

_{+}

*\ {*

_{1}*P*

_{∞}_{+}

*, P*

_{∞−}, P_{0}},(

*x, t*)

_{m}*∈ M, where M*⊆

*ℝ*(

^{2}is open and connected. Suppose u*x, t*)

_{m}*, v*(

*x, t*)

_{m}*∈ C*(

^{∞}*M*)

*satisfy the hierarchy of equations (2.7), and assume that λ*+

_{j}, 1 ≤ j ≤ 2 n*3, in (3.12) satisfy λ*\ {0}

_{j}∈ ℂ*, and λ*𝒟 μ ^ _ ( x , t m )
or equivalently,
𝒟 v _ ^ ( x , t m )
,

_{j}≠ λ_{k}as j ≠ k. Moreover, suppose that*is nonspecial for*(

*x, t*) ∈

_{m}*M. Then*

**Proof.** According to Riemann’s vanishing theorem [17,33], the definition and asymptotic properties of *ϕ*(*P*, *x*, *t _{m}*),

*ϕ*(

*P*,

*x*,

*t*) has expression of the following type

_{m}*N*(

*x*,

*t*) is independent of

_{m}*ϕ*(

*P*,

*x*,

*t*) near

_{m}*P*

_{∞ ±}and

*P*

_{0}, we have

## 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.

## References

### Cite this article

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 -