Algebro-Geometric Solutions of the Coupled Chaffee-Infante Reaction Diffusion Hierarchy

The coupled Chaffee-Infante reaction diffusion (CCIRD) hierarchy associated with a 
 
 3
 ×
 3
 
 matrix spectral problem is derived by using two sets of the Lenard recursion gradients. Based on the characteristic polynomial of the Lax matrix for the CCIRD hierarchy, we introduce a trigonal curve 
 
 
 
 K
 
 
 m
 −
 2
 
 
 
 of arithmetic genus 
 
 m
 −
 2
 
 , from which the corresponding Baker-Akhiezer function and meromorphic functions on 
 
 
 
 K
 
 
 m
 −
 2
 
 
 
 are constructed. Then, the CCIRD equations are decomposed into Dubrovin-type ordinary differential equations. Furthermore, the theory of the trigonal curve and the properties of the three kinds of Abel differentials are applied to obtain the explicit theta function representations of the Baker-Akhiezer function and the meromorphic functions. In particular, algebro-geometric solutions for the entire CCIRD hierarchy are obtained.


Introduction
It is significantly important to search for solutions of nonlinear partial differential equations of mathematical physics. There are many methods to find the exact solutions [1,2] and approximate solutions [1][2][3] of various nonlinear partial differential equations. Reaction diffusion equations are effective and important mathematical models, which contribute to explaining processes of the transition, diffusion, and fluidity of matter. Constructing exact solutions of such equations has been widely used in mathematics, physics, chemistry, biology, and other fields. Therefore, it is necessary for us to study algebro-geometric constructions of the coupled Chaffee-Infante reaction diffusion (CCIRD) hierarchy associated with a 3 × 3 matrix spectral problem. The third member in the hierarchy is u 1,t 2 = 12u 1,xx + 24 u 1 − u 2 1 u 2 À Á , which is called the CCIRD equation compared with Equation (0.4) in Ref. [4]. Algebro-geometric solution is closely associated with the inverse spectral theory [5,6], and the solution of the KdV equation with an initial value problem was solved by the use of the method in Ref. [7]. Over the recent decades, integrable equations related to 2 × 2 matrix spectral problems have been extensively researched. Several systematic methods have been developed to construct algebro-geometric solutions for integrable equations such as KdV, Kadomtsev-Petviashvili equation, modified KdV, sine-Gordon, Ablowitz-Kaup-Newell-Segur, the Camassa-Holm equations, and Ablowitz-Ladik lattice [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. But the study of algebrogeometric solutions of the whole hierarchy of 3 × 3 is still a challenging problem. Fortunately, in Ref. [28], a unified framework was proposed to yield algebro-geometric solutions of the whole Boussinesq hierarchy. Based on the work of that, a systematic method was proposed to define the trigonal curve and develop the framework to analyse soliton equations associated with the 3 × 3 matrix spectral problems, from which the algebro-geometric solutions of some entire hierarchies are obtained [29][30][31][32][33][34]. In Ref. [29], algebrogeometric quasi-periodic solutions to the three-wave resonant interaction hierarchy related to the trigonal curve with three infinite points were obtained. Wang and Geng constructed algebro-geometric solutions of a new hierarchy of soliton equations associated with a 3 × 3 matrix spectral problem [30] based on the methods used in [28,29]. Later, Ma analysed the four-component AKNS soliton hierarchy, particularly asymptotics of the Baker-Akhiezer functions, in such a way that it proposes a general theory applicable to soliton hierarchies associated with 3 × 3 matrix spectral problems [31]. As a continuous study of [31], Ma constructed algebro-geometric solutions of the four-component AKNS soliton hierarchy in terms of a general theory of trigonal curves [32]. However, as far as we know, algebro-geometric solutions to the CCIRD hierarchy have not been investigated. The most important result of this paper is to give the explicit algebro-geometric solutions to the CCIRD hierarchy related to 3 × 3 matrix spectral problems by using the approaches used in [28][29][30], which complements the existing works in this area.
The outline of this paper is as follows. In Section 2, we obtain the CCIRD hierarchy related to a 3 × 3 matrix spectral problem based on the Lenard recursion equations. In Section 3, a trigonal curve K m−2 of arithmetic genus m − 2 with three infinite points is introduced by the use of the characteristic polynomial of the Lax matrix for the stationary CCIRD equations, from which the stationary Baker-Akhiezer function and associated meromorphic functions are given on K m−2 . Then, the stationary CCIRD equations are decomposed into the system of Dubrovin-type ordinary differential equations. In Section 4, we present the explicit theta function representations of the stationary Baker-Akhiezer function, of the meromorphic functions, and, in particular, of the potentials for the entire stationary CCIRD hierarchy. In Section 5, we extend all the Baker-Akhiezer functions, the meromorphic functions, the Dubrovin-type equations, and the theta function representations dealt with in Sections 3 and 4 to the time-dependent case.

The CCIRD Hierarchy
In the section, we shall derive the CCIRD hierarchy associated with a 3 × 3 spectral problem: 1 1 where the potential u = ðu 1 , u 2 Þ T and λ is a spectral parameter. Next, we introduce the Lenard gradient sequences Kg j−1 = Jg j , g j u i =0 = 0, j ≥ 0, with two initial points g −1 = 0, 0, 0, 0, 0, 0, 1, 0 ð Þ T , and two operators are defined as Then, the sequences fg j g and fĝ j g, j ≥ 0, can be uniquely determined and the first several members read as

The Stationary Baker-Akhiezer Function
In the section, we are devoted to detailed study of the stationary Baker-Akhiezer function and the associated meromorphic functions. Then, the system of Dubrovin-type differential equations is derived. Let us consider the stationary CCIRD hierarchy: X n = 0, n ≥ 0, which is equivalent to the stationary zero-curvature equation, V ðnÞ x = ½U, V ðnÞ , V ðnÞ = ðλ n VÞ + = ðV ðnÞ i,j Þ 3×3 , V ðnÞ i,j = ∑ n k=0 V ij,k−1 λ n−k , and V ij,k−1 determined by (21). It is easy to verify that the matrix yI − V ðnÞ also satisfies the stationary zero-curvature equation. Then, the characteristic polynomial of Lax matrix V ðnÞ , F m ðλ, yÞ = det ðyI − V ðnÞ Þ, is independent of variable x with the expansion 3 Advances in Mathematical Physics where S m ðλÞ and T m ðλÞ are polynomials with constant coefficients of λ: It is easy to find that T m ðλÞ is a polynomial of degree 3n with respect to λ as α 0 β 0 ðα 0 + β 0 Þ ≠ 0. Then, F m ðλ, yÞ = 0 naturally leads to a trigonal curve with m = 3n. For convenience, we denote the compactification of the curve K m−2 by the same symbol K m−2 . Hence, K m−2 becomes a three-sheeted Riemann surface of arithmetic genus m − 2 if it is nonsingular or smooth. Here, the meaning of nonsingular is that at each point P ′ = ðλ ′ , y ′ Þ ∈ K m−2 , ð∂F m ðλ, yÞ/∂λ, ∂F m ðλ, yÞ/∂yÞj ðλ,yÞ=ðλ ′ ,y ′Þ ≠ 0 holds. For m ≥ 4, these curves are typically nonhyperelliptic. Point P on K m−2 is represented as pairs P = ðλ, yðPÞÞ satisfying (29) along with P ∞ j = ð∞,∞ j Þ, j = 1, 2, 3, the three different points at infinity, which can be computed from the curve y 3 + S m ðλÞy − T m ðλÞ = 0 by choosing λ = ζ −1 . The complex structure on K m−2 is defined in the usual way by introducing local coordinate ζ Q = λ − λ Q near point Q = ðλ Q , yðQÞÞ ∈ K m−2 which is neither branch nor singular point of K m−2 except the three points P ∞ 1 , P ∞ 2 , P ∞ 3 , at infinity with local coordinate λ = ζ −1 and local coordinate Next, we shall define the meromorphic functions ϕ 2 ðP, xÞ and ϕ 3 ðP, xÞ on K m−2 as follows: with the stationary Baker-Akhiezer function ψðP, x, x 0 Þ defined by By using (31)-(33), a direct calculation gives that where Through straightforward calculations, we obtain some main interrelationships among polynomials A m , ⋯, Advances in Mathematical Physics By observing Equations (21) and (38), one infers that E m−2 , F m−2 , and F m−2 are polynomials with respect to λ of degree m − 2. Therefore, we may write them in the following forms: where In order to more distinctly put forward the properties of ϕ 2 ðP, xÞ, ϕ 3 ðP, xÞ, and ψ 1 ðP, x, x 0 Þ, we introduce the holomorphic map * , changing sheets, which is defined by * :

Algebro-Geometric Solutions of the Stationary CCIRD Hierarchy
In the section, we continue our study of the stationary CCIRD hierarchy and will obtain explicit Riemann theta function representations for the two meromorphic functions ϕ 2 ðP, xÞ, ϕ 3 ðP, xÞ, the Baker-Akhiezer function ψ 1 ðP, x, x 0 Þ, and the algebro-geometric solutions u 1 and u 2 for the CCIRD hierarchy. By introducing the local coordinate ζ = λ −1 near P ∞ 1 , P ∞ 2 , P ∞ 3 ∈ K m−2 , we have the following lemma.

Lemma 2.
Suppose that u satisfies the nth stationary CCIRD system X n = 0.
Proof. Substituting the three sets of ansatz  A tedious calculation reveals that the asymptotic behaviors of yðPÞ and S m near P ∞ 1 , P ∞ 2 , P ∞ 3 are given as Equip the Riemann surface K m−2 with an appropriate fixed homology basis fa j , b j g m−2 j=1 , in such a way that the intersection matrix of cycles satisfies a j ∘ b k = δ j,k , a j ∘ a k = 0, b j ∘ b k = 0, j, k = 1, ⋯, m − 2: For the present, we introduce the holomorphic differentials ϖ l ðPÞ on K m−2 defined by By using the basis a j and b j , the matrices A and B can be constructed from and it is possible to show that matrices A and B are invertible. Now, we define the matrices C and τ by C = A −1 , τ = A −1 B: One can see that matrix τ = ðτ ij Þ ðm−1Þ×ðm−1Þ is symmetric, and it has a positive definite imaginary part. If we normalize ϖ l into the new basis ω j , Then, the Laurent expansion of (83) near fP ∞ 1 , P ∞ 2 , P ∞ 3 g yields the following results: Let ω ð2Þ P ∞ s ,2 ðPÞ, s = 1, 2, 3, denote the normalized Abelian differential of the second kind satisfying and introduce then we have where e 3 ðQ 0 Þ are integral constants and Q 0 is an appropriately chosen base point on K m−2 \ fP ∞ 1 , P ∞ 2 , P ∞ 3 g. The b-periods of the differential Ω ð2Þ ðPÞ are denoted by

Advances in Mathematical Physics
Then, from (85) and (87), we have in which we used Furthermore, the normalized Abelian differential of the third kind ω ð3Þ P ∞ 3 ,P ∞ j ðPÞ, j = 1, 2, is holomorphic on K m−2 \ f P ∞ 3 , P ∞ j g with simple poles at P ∞ 3 and P ∞ j with residues 1 and -1, respectively, that is, where e ð3Þ k,∞ j ðQ 0 Þ, k = 1, 2, j = 1, 2, 3, are integration constants. Let T m−2 be the period lattice fZ ∈ ℂ m−2 | Z = N + τL ; N , L ∈ ℤ m−2 g. The complex torus J m−2 = ℂ m−2 /T m−2 is called the Jacobian variety of K m−2 . An Abel map A : with the natural linear extension to the factor group DivðK m−2 Þ A 〠n k P k = 〠n k A P k ð Þ: ð96Þ where P Theorem 3. Assume that the curve K m−2 is nonsingular, and let x, x 0 ∈ ℂ. Then, Proof. We prove only the first linearity of the Abel map with respect to x in (98). Assume that μ j ðxÞ ≠ μ j ′ ðxÞ for j ≠ j ′ ; then, one computes 9 Advances in Mathematical Physics which yields by the use of the standard Lagrange interpolation argument that which implies the first representation of (98). The second and third equalities in (98) follow from the same calculation. Denote by θðzÞ the Riemann theta function associated with K m−2 equipped with a fixed homology basis. For convenience, the function z : where M is the vector of Riemann constants. Then, we get In view of (98), we could rewrite them as where M ðjÞ 2 x 0 , j = 1, 2, 3, s = 1, 2, 3: Combined with the above results, the theta function representations of ϕ 2 ðP, xÞ, ϕ 3 ðP, xÞ, ψ 1 ðP, x, x 0 Þ and the algebro-geometric solutions of the stationary CCIRD hierarchy are presented in the next theorem.

Algebro-Geometric Solutions of the CCIRD Hierarchy
In this section, we extend the results of Sections 3 and 4 to the time-dependent CCIRD hierarchy. In particular, we obtain Riemann theta function representations for the timedependent Baker-Akhiezer function, the meromorphic function, and algebro-geometric solutions of the CCIRD hierarchy. Similar to (31)-(33), we consider the following timedependent Baker-Akhiezer function: x, x 0 , t r , t 0,r ð Þ = y P ð Þψ P, x, x 0 , t r , t 0,r ð Þ , ψ 1 P, x 0 , x 0 , t 0,r , t 0,r ð Þ = 1, The compatibility conditions of the first three equations in (116) show that It is easy to find that yI − V ðnÞ satisfies (118) and (119). Then, the characteristic polynomial of Lax matrix V ðnÞ for the CCIRD hierarchy is a constant independent of variables x and t r with the expansion 11 Advances in Mathematical Physics where S m ðλÞ and T m ðλÞ are defined as in (27) and (28). Then, the CCIRD curve K m−2 is determined by Closely related to ψðP, x, x 0 , t r , t 0,r Þ are the following two meromorphic functions ϕ 2 ðP, x, t r Þ and ϕ 3 ðP, x, t r Þ on K m−2 defined by which imply from (116) that where P = ðλ, yÞ ∈ K m−2 , ðx, t r Þ ∈ ℂ 2 and A m ðλ, x, t r Þ, ⋯, After defining b μ j ðx, t r Þ, b ν j ðx, t r Þ, b ξ j ðx, t r Þ as (46)-(48) by replacing ðxÞ with ðx, t r Þ, one infers from (74), (75), (124), and (128) that the divisors ðϕ 2 ðP, x, t r ÞÞ and ðϕ 3 ðP, x, t r ÞÞ of ϕ 2 ðP, x, t r Þ and ϕ 3 ðP, x, t r Þ are as follows: Differentiating (122) and (123) with respect to t r and using (116), we get Further properties of ϕ 2 ðP, x, t r Þ and ϕ 3 ðP, x, t r Þ can be presented, similar to (55)-(63), replacing ðxÞ with ðx, t r Þ, ðP, xÞ with ðP, x, t r Þ, etc. The four important ones of that are given as follows: Lemma 5. Assume (116) and (117) and let ðλ, x, t r Þ ∈ ℂ 3 . Then, Thus, we prove the expression (133). The last one can be proved in the same way.
We present some properties of ψ 1 ðP, x, x 0 , t r , t 0,r Þ as follows.
Let ω ð2Þ P ∞ l ,j , j ∈ ℕ, l = 1, 2, 3, be the normalized differential of the second kind holomorphic on K m−2 \ fP ∞ l g with a pole of order j at P ∞ l , Furthermore, we define the normalized differential of the second kind bỹ In addition, we define the vector of b-periods of the differential of the second kindΩ