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]

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]

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]

where u and v are two potentials, and λ is a constant spectral parameter. To this end, we first introduce the Lenard recursion sequences with three starting points where K and J are two operators defined by

It is easy to see that KerJ={α0L0+α¯0L¯0+α˜0L˜0∣∀α0,α¯0,α˜0∈𝕉} . Hence L_j 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

where V11(m),V12(m),V21(m) are defined as follows;

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

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

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

which satisfies the Lax equations

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

and

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

where

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

where E_j = (c_j, d_j)^T, 0 ≤ j ≤ n − 1. It is easy to see that the equation JE₀ = 0 has a special general solution

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

which are special solutions of (3.7), where α₁, α₂,…,α_k are constants of integration and α₀ = 1. Moreover, from (3.8) we can get 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 t_m, we have

one is naturally led to introduce the hyperelliptic curve 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_j of 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

where {μj}j=1n+1 and {vj}j=1n+1 are called elliptic variables. According to the definition of 𝒦_{n + 1}, we can lift the roots μ_j and v_j to 𝒦_{n + 1} by introducing where (x, t_m) ∈ ℝ².

From the following lemma, we can explicitly represent α_l (0 ≤ l ≤ n) by the constants λ₁,…,λ_{2 n + 3}.

4. Finite genus solutions

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

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

which are 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⁻¹B. The matrix τ can be shown to be symmetric (τ_kj = τ_jk), and it has positive definite imaginary part (Imτ > 0). If we normalize w˜l into the new basis ω_j,

then we obtain

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

where P,Pk∈𝒦n+1,ω_=(ω1,…,ωn+1) .

Let θ(z) denote the Riemann theta function associated with 𝒦_{n + 1} [33–35]:

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 where P ∈ 𝒦_{n + 1}, Q = {Q₁,⋯,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 Λ_=(Λ1,…,Λn+1) is the vector of Riemann constant defined by

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, t_m) on 𝒦_{n + 1}:

where P = (λ, y) ∈ 𝒦_{n + 1} \ {P_{∞ ±}}.

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

from the Lemma 4.1 and the definition of ϕ(P, x, t_m) in (4.10).

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

where γ_i ∈ ℂ, j = 1, …, n + 1, are constants that are determined by

The explicit formula (4.21) then implies

Therefore,

for some constants ω^{∞ +}, ω^∞−, ω⁰ ∈ ℂ.