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 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.
- 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]
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 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]
It is easy to see that
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 = (χ1, χ2)T and ψ = (ψ1, ψ2)T be two basic solutions of (2.1) and (2.5). We introduce a Lax matrix
Therefore, detW is a constant independent of x and tm. 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
By induction, we obtain from recursive relations (3.7) and (2.2) that
Since detW is a (2n + 4) th-order polynomial in λ, whose coefficients are constants independent of x and tm, we have
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
From the following lemma, we can explicitly represent αl (0 ≤ l ≤ n) by the constants λ1,…,λ2 n + 3.
Lemma 3.1.
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 λn + 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
Hence,
Considering
Therefore, we compute that
4. Finite genus solutions
Equip the 𝒦n + 1 with canonical basis cycles:
For the present, we will choose our basis as the following set [17]
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
Let 𝒯n + 1 be the period lattice
Let θ(z) denote the Riemann theta function associated with 𝒦n + 1 [33–35]:
Without loss of generality, we choose the branch point
By virtue of (3.12) and (3.13) we can define the meromorphic function ϕ(P, x, tm) on 𝒦n + 1:
Lemma 4.1.
Suppose that u(x, tm), v(x, tm) ∈ C∞(ℝ2) satisfy the hierarchy (2.7). Let λj ∈ ℂ \{0}, 1 ≤ j ≤ 2n + 3, and P = (λ, y) ∈ 𝒦n + 1 \ {P∞ +, P∞ +, P0}, where, P0 = (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, tm) in (4.10), we have
To prove (4.12), we introduce the local coordinate
The divisor of ϕ(P, x, tm) is given by
Let
The explicit formula (4.21) then implies
Therefore,
Theorem 4.1.
Let P = (λ, y) ∈ 𝒦n + 1 \ {P∞ +, P∞−, P0},(x, tm) ∈ M, where M ⊆ ℝ2 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
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
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 -