{'fraction_non_alphanumeric': 0.07020069808027923, 'fraction_numerical': 0.02988656195462478, 'mean_word_length': 3.9980375054513737, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 8, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Algebro-geometric solutions for the two-component Hunter-Saxton hierarchy

This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the two-component Hunter-Saxton (HS2) hierarchy through an algebro-geometric initial value problem. Our main tools include the polynomial recursive formalism, the hyperelliptic curve with finite number of genus, the Baker-Akhiezer functions, the meromorphic function, the Dubrovin-type equations for auxiliary divisors, and the associated trace formulas. With the help of these tools, the explicit representations of the algebro-geometric solutions are obtained for the entire HS2 hierarchy.


Introduction
In this paper, we consider the following two-component Hunter-Saxton (HS2) system { m t + 2u x m + um x + σ ρρ x = 0, ρ t + (uρ) x = 0, (1.1) where m = −u xx , σ = ±1. The system (1.1) was recently introduced by Constantin and Ivanov in [9]. The variable u(x,t) describes the horizontal velocity of the fluid and the variable ρ(x,t) is in connection with the horizontal deviation of the surface from equilibrium, all measured in dimensionless units [9]. The HS2 system (1.1) is the short-wave (or high-frequency) limit of the two-component Camassa-Holm shallow water system [9,17,39,43]. This system, reading as (1.1) with m replaced by (1 − ∂ 2 xx )u, has recently been the object of intensive study (see [8,9,25,26,45]). The HS2 system (1.1) is integrable, since it has a Lax pair [9] and a bi-Hamiltonian structure [9,39]. It is also a particular case of the Gurevich-Zybin system describing the dynamics in a model of nondissipative dark matter [24,40,44]. The mathematical properties of the system (1.1) have been studied further in a series of works (see [9,23,24,36,[42][43][44]). For ρ = 0, the system (1.1) becomes the Hunter-Saxton (HS) equation [30], which models the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field. Here, u(x,t) describes the director field of a nematic liquid crystal, x is a space variable in a reference frame moving with the linearized wave velocity, and t is a slow time variable. The field of unit vectors (cos u(x,t), sin u(x,t)) describes the orientation of the molecules [23,30,44].
The HS equation also arises in a different physical context as the high-frequency limit [13,31] of the Camassa-Holm (CH) equation -a model equation for shallow water waves [7,11,32] and a re-expression of the geodesic flow on the diffeomorphism group of the circle [10] with a bi-Hamiltonian structure [18], which is completely integrable [12]. The HS equation is also a completely integrable system with a bi-Hamiltonian structure, and hence, it possesses a Lax pair, an infinite family of commuting Hamiltonian flows, as well as an associated sequence of conservation laws [5,30,31,39,41]. It also describes the geodesic flow on the homogeneous space related to the Virasoro group [33][34][35]. Recently, algebro-geometric solutions for the HS hierarchy was investigated in [28]. Moreover, peakon solutions, global weak solutions and the Cauchy problem of the system (1.1) were discussed in [9,23,36,44]. However, within the knowledge of the authors, the algebro-geometric solutions of the entire HS2 hierarchy are not studied yet.
This paper concerns algebro-geometric quasi-periodic solutions of the whole HS2 hierarchy associated with (1.1). The algebro-geometric solution, as an important feature of integrable system, is a kind of explicit solution closely related to the inverse spectral theory [1,6,[14][15][16]. In a degenerated case of the algebro-geometric solution, the multi-soliton solution and periodic solution in elliptic function type may be obtained [6,37,38]. A systematic approach, proposed by Gesztesy and Holden to construct algebro-geometric solutions for integrable equations, has been extended to the whole (1+1) dimensional integrable hierarchy, such as the AKNS hierarchy, the CH hierarchy, etc. [19][20][21][22]. Recently, we investigated algebro-geometric solutions for the modified CH hierarchy and the Degasperis-Procesi hierarchy [27,29].
The outline of the present paper is as follows.
In section 2, based on the polynomial recursion formalism, we derive the HS2 hierarchy associated with the 2 × 2 spectral problem. A hyperelliptic curve K n of arithmetic genus n is introduced with the help of the characteristic polynomial of Lax matrix V n for the stationary HS2 hierarchy.
In Section 3, we decompose the stationary HS2 equations into a system of Dubrovin-type equations. Moreover, we obtain the stationary trace formulas for the HS2 hierarchy.
In Section 4, we present the first set of our results, the explicit theta function representations of the potentials u, ρ for the entire stationary HS2 hierarchy. Furthermore, we study the initial value problem on an algebro-geometric curve for the stationary HS2 hierarchy.
In Sections 5 and 6, we extend the analyses of Sections 3 and 4, respectively, to the timedependent case. Each equation in the HS2 hierarchy is permitted to evolve in terms of an independent time parameter t r . As initial data, we use a stationary solution of the nth equation and then construct a time-dependent solution of the rth equation of the HS2 hierarchy. The Baker-Akhiezer function, the analogs of the Dubrovin-type equations, the trace formulas, and the theta function representations in Section 4 are all extended to the time-dependent case.
Finally, we remark that although our focus in this paper is on the case σ = 1 in (1.1), all of the arguments presented here can be adapted to study the corresponding case σ = −1.

The HS2 hierarchy
In this section, we provide the construction of HS2 hierarchy and derive the corresponding sequence of zero-curvature pairs using a polynomial recursion formalism. Moreover, we introduce the underlying hyperelliptic curve in connection with the stationary HS2 hierarchy.
Throughout this section, we make the following hypothesis.
Next, it is convenient to introduce the corresponding homogeneous coefficientsf l ,ĝ l , andĥ l , defined by the vanishing of the integration constants c k , k = 1, . . . , l, (2.7) Hence, Now, given Hypothesis 2.1, one introduces the following 2 × 2 matrix U by ψ, (2.10) and for each n ∈ N 0 , the following 2 × 2 matrix V n by ψ t n = V n (z)ψ, assuming F n , G n , and H n to be polynomials a with respect to z and C ∞ in x. The compatibility condition of linear system (2.10) and (2.11) yields the stationary zero-curvature equation which is equivalent to 14) and hence where the polynomial R 2n+2 of degree 2n + 2 is x-independent. In another way, one can write R 2n+2 as Here, we emphasize that the coefficient ( (2.20) Then comparing the coefficient of powers z 2n+2 yields Therefore, For simplicity, we denote it by a 2 , a ∈ C. Then, R 2n+2 (z) can be rewritten as Next, we compute the characteristic polynomial det(yI − zV n ) of Lax matrix zV n , and then introduce the (possibly singular) hyperelliptic curve K n of arithmetic genus n defined by In the following, we will occasionally impose further constraints on the zeros E m of R 2n+2 introduced in (2.24) and assume that The stationary zero-curvature equation (2.13) implies polynomial recursion relations (2.3). Introducing the following polynomials F n (z), G n (z), and H n (z) with respect to the spectral parameter z, 3) for f l , l = 0, . . . , n + 1, and g l , l = 0, . . . , n. For fixed n ∈ N 0 , we obtain the recursion relations for h l , l = 0, . . . , n − 1 in (2.3) and Moreover, from (2.15), one infers that Then using (2.31) and (2.32) permits one to write the stationary HS2 hierarchy as We record the first equation explicitly, By definition, the set of solutions of (2.33) represents the class of algebro-geometric HS2 solutions, with n ranging in N 0 and c l in C, l ∈ N. We call the stationary algebro-geometric HS2 solutions u, ρ as HS2 potentials at times.

Remark 2.2.
Here, we emphasize that if u, ρ satisfy one of the stationary HS2 equations in (2.33) for a particular value of n, then they satisfy infinitely many such equations of order higher than n for certain choices of integration constants c l . This is a common characteristic of the general integrable soliton equations such as the KdV, AKNS, and CH hierarchies [21].
Next, we introduce the corresponding homogeneous polynomials F l , G l , H l by In accordance with our notation introduced in (2.7) and (2.35)-(2.39), the corresponding homogeneous stationary HS2 equations are then defined by s-HS2 n (u, ρ) = s-HS2 n (u, ρ)| c l =0, l=1,...,n = 0, n ∈ N 0 . (2.40) At the end of this section, we turn to the time-dependent HS2 hierarchy. In this case, u, ρ are considered as functions of both space and time. We introduce a deformation parameter t n ∈ R in u and ρ, replacing u(x), ρ(x) by u(x,t n ), ρ(x,t n ), for each equation in the hierarchy. In addition, the definitions (2.10), (2.12), and (2.28)-(2.30) of U, V n and F n , G n , and H n , respectively, still apply. The corresponding zero-curvature equation reads which results in the following set of equations For fixed n ∈ N 0 , inserting the polynomial expressions for F n , G n , and H n into (2.42)-(2.44), respectively, first yields recursion relations (2. 3) for f l | l=0,...,n+1 , g l | l=0,...,n , h l | l=0,...,n−1 and Moreover, using (2.44), one finds Hence, using (2.45) and (2.46) permits one to write the time-dependent HS2 hierarchy as For convenience, we record the first equation in this hierarchy explicitly, The first equation HS2 0 (u, ρ) = 0 (with c 1 = 0) in the hierarchy represents the HS2 system as discussed in section 1. Similarly, one can introduce the corresponding homogeneous HS2 hierarchy by HS2 n (u, ρ) = HS2 n (u, ρ)| c l =0, l=1,...,n = 0, n ∈ N 0 . (2.49) In fact, since the Lenard recursion formalism is almost universally adopted in the contemporary literature, we thought it might be worthwhile to use the Gesztesy and Holden's method, the polynomial recursion formalism, to construct the HS2 hierarchy.

The stationary HS2 formalism
This section is devoted to a detailed study of the stationary HS2 hierarchy. We first define a fundamental meromorphic function ϕ (P, x) on the hyperelliptic curve K n , using the polynomial recursion formalism described in section 2, and then study the properties of the Baker-Akhiezer function ψ(P, x, x 0 ), Dubrovin-type equations, and trace formulas.
For major parts of this section, we assume ( which is compactified by joining two points at infinity P ∞ ± , with P ∞ + ̸ = P ∞ − . But for notational simplicity, the compactification is also denoted by K n . Hence, K n becomes a two-sheeted Riemann surface of arithmetic genus n. Points P on K n \{P ∞± } are denoted by P = (z, y(P)), where y(·) is the meromorphic function on K n satisfying F n (z, y(P)) = 0. The complex structure on K n is defined in the usual way by introducing local coordinates near points Q 0 = (z 0 , y(Q 0 )) ∈ K n , which are neither branch nor singular points of K n ; near the branch and singular points Q 1 = (z 1 , y(Q 1 )) ∈ K n , the local coordinates are near the points P ∞ ± ∈ K n , the local coordinates are The holomorphic map * , changing sheets, is defined by * : where y j (z), j = 0, 1 denote the two branches of y(P) satisfying F n (z, y) = 0, namely, Taking into account (3.3), one easily finds y 0 + y 1 = 0, Moreover, positive divisors on K n of degree n are denoted by D P 1 ,...,P n : (3.5) Next, we define the stationary Baker-Akhiezer function ψ(P, (3.6) Closely related to ψ(P, x, x 0 ) is the following meromorphic function ϕ (P, x) on K n defined by such that Then, based on (3.6) and (3.7), a direct calculation shows that , (3.9) and In the following, the roots of polynomials F n and H n will play a special role, and hence, we Due to assumption (2.1), u and ρ are smooth and bounded, and hence, F n (z, x) and H n (z, x) share the same property. Thus, one concludes taking multiplicities (and appropriate reordering) of the zeros of F n and H n into account. From (3.9), the divisor (ϕ (P, x)) of ϕ (P, x) is given by Here, we abbreviatedμ Further properties of ϕ (P, x) are summarized as follows.
as well as Proof. Equation The properties of ψ(P, x, x 0 ) are summarized in the following lemma.
Proof. Equation (3.21) is a consequence of (2.14), (3. In Lemma 3.2, we denote by which is equivalent to the basic identity (2.18), z 2 G 2 n + F n H n = R 2n+2 . This fact reveals the relations between our approach and the algebro-geometric solutions of the HS2 hierarchy.
Remark 3.3. The Baker-Akhiezer function ψ of the stationary HS2 hierarchy is formally analogous to that defined in the context of KdV or AKNS hierarchies. However, its actual properties in a neighborhood of its essential singularity will feature characteristic differences to standard Baker-Akhiezer functions (cf. Remark 4.2).
Next, we derive Dubrovin-type equations, that is, first-order coupled systems of differential equations that govern the dynamics of µ j (x) and ν l (x) with respect to variations of x.
..,n satisfy the system of differential equations,

27)
with initial conditions

30)
with initial conditions

Co-published by Atlantis Press and Taylor & Francis Copyright: the authors
Proof. It suffices to prove (3.27) and (3.29) since the proof of (3.30) and (3.32) follow in an identical manner. Differentiating (3.11) with respect to x then yields On the other hand, taking into account equation (2.14), one finds Then combining equation (3.33) with (3.34) leads to (3.27). The proof of smoothness assertion (3.29) is analogous to the KdV case in [21].
Next, we turn to the trace formulas of the HS2 invariants, that is, expressions of f l and h l in terms of symmetric functions of the zeros µ j and ν l of F n and H n , respectively. For simplicity, we just record the simplest case.
Proof. Equation (3.35) follows by considering the coefficient of z n in F n in (2.28) and (3.11), which yields The constant c 1 can be determined by a long straightforward calculation considering the coefficient of z 2n+1 in (2.18), which results in

Stationary algebro-geometric solutions of HS2 hierarchy
In this section, we obtain explicit Riemann theta function representations for the meromorphic function ϕ , and especially, for the solutions u, ρ of the stationary HS2 hierarchy. We begin with the asymptotic properties of ϕ and ψ j , j = 1, 2.
Proof. The existence of the asymptotic expansions of ϕ in terms of the appropriate local coordinates ζ = z −1 near P ∞ ± and ζ = z near P 0 is clear from its explicit expression in (3.9). Next, we compute the coefficients of these expansions utilizing the Riccati-type equation (3.17). Indeed, inserting the ansatz into (3.17) and comparing the same powers of z then yields (4.1). Similarly, inserting the ansatz into ( Remark 4.2. We note the unusual fact that P 0 , as opposed to P ∞± , is the essential singularity of ψ j , j = 1, 2. In addition, one easily finds the leading-order exponential term in ψ j , j = 1, 2, near P 0 is x-dependent, which makes matters worse. This is in sharp contrast to standard Baker-Akhiezer functions that typically feature a linear behavior with respect to x in connection with their essential singularities of the type exp(c(x − x 0 )ζ −1 ) near ζ = 0.
Next, we introduce the holomorphic differentials η l (P) on K n η l (P) = a z l−1 y(P) dz, l = 1, . . . , n, (4.9) and choose a homology basis {a j , b j } n j=1 on K n in such a way that the intersection matrix of the cycles satisfies Associated with K n , one introduces an invertible matrix E ∈ GL(n, C) (4.11) Apparently, the matrix τ is symmetric and has a positive-definite imaginary part.
We choose a fixed base point Q 0 ∈ K n \ {μ 0 (x),ν 0 (x)}. The Abel maps A Q 0 (·) and α Q 0 (·) are defined by and Here The following result shows the nonlinearity of the Abel map with respect to the variable x, which indicates a characteristic difference between the HS2 hierarchy and other completely integrable systems such as the KdV and AKNS hierarchies.
In particular, the Abel map does not linearize the divisor Dμ 0 (x)μ(x) on Ω µ .
The fact that the Abel map does not provide the proper change of variables to linearize the divisor Dμ 0 (x)μ(x) in the HS2 context is in sharp contrast to standard integrable soliton equations such as the KdV and AKNS hierarchies. However, the change of variables linearizes the Abel map A Q 0 (Dˆμ 0 (x)μ(x) ),μ j (x) = µ j (x), j = 0, . . . , n. The intricate relation between the variable x andx is detailed in (4.34).
for some constants e 0 , d 0 ∈ C. We also record In the following, it will be convenient to introduce the abbreviations where Ξ Q 0 is the vector of Riemann constants (cf.(A.45) [21]). It turns out that z(·, Q) is independent of the choice of base point Q 0 (cf.(A.52), (A.53) [21]).
Based on above preparations, we will give explicit representations for the meromorphic function ϕ and the stationary HS2 solutions u, ρ in terms of the Riemann theta function associated with K n .
Next, we derive the representations of ϕ , u, and ρ in terms of the Riemann theta function. A special case of Riemann's vanishing theorem (cf. Theorem A.26 [21]) yields (4.37) Therefore, the divisor (3.15) shows that ϕ (P, x) has expression of the type where C(x) is independent of P ∈ K n . Then taking into account the asymptotic expansion of ϕ (P, x) near P 0 in (4.2), we obtain (4.31). The representation (4.32) for u on Ω follows from trace formula (3.35) and the expression (F.59) [21] for ∑ n j=0 µ j . The representation (4.33) for ρ on Ω is clear from  Remark 4.7. Since by (3.15) Dν 0ν and Dμ 0μ are linearly equivalent, that is Then one infers Hence, one can eliminate Dν (x) in (4.31), in terms of Dμ (x) using z(P,ν) = z(P,μ) + ∆, P ∈ K n . (4.41) Remark 4.8. We emphasized in Remark 4.2 that ψ in (3.8) and (3.10) differs from standard Baker-Akhiezer functions. Hence, one can not expect the usual theta function representation of ψ j , j = 1, 2, in terms of ratios of theta functions times an exponential term containing a meromorphic differential with a pole at the essential singularity of ψ j multiplied by (x − x 0 ). However, using (E.3) and (F.59) [21], one computes and hence obtains the theta function representation of ψ 1 upon inserting (4.42) into (3.21). Then, the corresponding theta function representation of ψ 2 follows by (3.10) and (4.31).

The time-dependent HS2 formalism
In this section, we extend the algebro-geometric analysis of Section 3 to the time-dependent HS2 hierarchy. Throughout this section, we assume (2.2) holds. The time-dependent algebro-geometric initial value problem of the HS2 hierarchy is to solve the time-dependent rth HS2 flow with a stationary solution of the nth equation as initial data in the hierarchy. More precisely, given n ∈ N 0 , based on the solution u (0) , ρ (0) of the nth stationary HS2 equation s-HS2 n (u (0) , ρ (0) ) = 0 associated with K n and a set of integration constants {c l } l=1,...,n ⊂ C, we want to construct a solution u, ρ of the rth HS2 flow HS2 r (u, ρ) = 0 such that u(t 0,r ) = u (0) , ρ(t 0,r ) = ρ (0) , for some t 0,r ∈ R, r ∈ N 0 .
To emphasize that the integration constants in the definitions of the stationary and the timedependent HS2 equations are independent of each other, we indicate this by adding a tilde on all the time-dependent quantities. Hence, we employ the notation V r , F r , G r , H r ,f s ,g s ,h s ,c s in order to distinguish them from V n , F n , G n , H n , f l , g l , h l , c l in the following. In addition, we mark the individual rth HS2 flow by a separate time variable t r ∈ R.

(5.50)
Proof. To prove (5.46), we first consider the part of time variable in the definition (5.23), that is, exp ) . (5.51) The integrand in the above integral equals On the other hand, the part of space variable in (5.23) can be written as In analogy to Lemma 3.4, the dynamics of the zeros {µ j (x,t r )} j=0,...,n and {ν l (x,t r )} l=0,...,n of F n (z, x,t r ) and H n (z, x,t r ) with respect to x and t r are described in terms of the following Dubrovintype equations. (i) Suppose that the zeros {µ j (x,t r )} j=0,...,n of F n (z, x,t r ) remain distinct for (x,t r ) ∈ Ω µ , where Ω µ ⊆ R 2 is open and connected, then {µ j (x,t r )} j=0,...,n satisfy the system of differential equations,  On the other hand, inserting z = µ j into (5.38) and using (5.26), one finds Since the stationary trace formulas for HS2 invariants in terms of symmetric functions of µ j in Lemma 3.5 extend line by line to the corresponding time-dependent setting, we next record the t r -dependent trace formulas without proof. For simplicity, we confine ourselves to the simplest one only.

Time-dependent algebro-geometric solutions of HS2 hierarchy
In our final section, we extend the results of section 4 from the stationary HS2 hierarchy, to the timedependent case. We obtain Riemann theta function representations for the meromorphic function ϕ , and especially, for the algebro-geometric solutions u, ρ of the whole HS2 hierarchy. We first record the asymptotic properties of ϕ in the time-dependent case.
Since the proof of Lemma 6.1 is identical to the corresponding stationary results in Lemma 4.1, we omit the corresponding details.

Co-published by Atlantis Press and Taylor & Francis
Copyright: the authors + 4a which is equivalent to (6.8).
The analogous results hold for the corresponding divisor Dν 0 (x,t r )ν(x,t r ) associated with ϕ (P, x,t r ).
Next, recalling the definition of B Q 0 andβ Q 0 in (4.19) and (4.20), one obtains the following result.
The fact that the Abel map does not effect a linearization of the divisor Dμ 0 (x,t r )μ(x,t r ) in the time-dependent HS2 context, which is well known and discussed (using different approaches) by Constantin and McKean [12], Alber, Camassa, Fedorov, Holm, and Marsden [2], Alber and Fedorov [3,4]. The change of variables and

Co-published by Atlantis Press and Taylor & Francis
Copyright: the authors − 4a linearizes the Abel map A Q 0 (Dˆμ 0 (x,t r )μ(x,t r ) ),μ j (x,t r ) = µ j (x,t r ), j = 0, . . . , n. The intricate relation between the variables (x,t r ) and (x,t r ) is detailed in (6.21). Our approach follows a route similar to Gesztesy and Holden's treatment of the CH hierarchy [21].
Next, we shall provide the explicit representations of ϕ and u, ρ in terms of the Riemann theta function associated with K n , assuming the affine part of K n to be nonsingular. Recalling (4.24)-(4.30), the analog of Theorem 4.5 in the stationary case then reads as follows.
Remark 6.6. Remark 4.7 applies in the present time-dependent context. Moreover, to obtain the theta function representation of ψ j , j = 1, 2,, one can write F r in terms of Ψ k (μ) and use (5.46), in analogy to the stationary case discussed in Remark 4.8. Here we omit further details.