## Abstract

We provide a detailed treatment of real-valued, smooth and bounded algebro-geometric solutions of the Camassa–Holm (CH) hierarchy and describe the associated isospectral torus. We employ Dubrovin-type equations for auxiliary divisors and certain aspects of direct and inverse spectral theory for self-adjoint Hamiltonian systems. In particular, we rely on Weyl–Titchmarsh theory for singular (canonical) Hamiltonian systems. We also briefly discuss real-valued algebro-geometric solutions with a cusp behaviour. While we focus primarily on the case of stationary algebro-geometric CH solutions, we note that the time-dependent case subordinates to the stationary one with respect to isospectral torus questions.

## 1. Introduction

In Gesztesy & Holden (2002) we provided a detailed treatment of the Camassa–Holm (CH) hierarchy with special emphasis on its algebro-geometric solutions. The first nonlinear partial differential equation of this hierarchy, the Camassa–Holm equation, also known as the dispersive shallow water equation (Camassa & Holm 1993) is given by(1.1)(choosing a convenient scaling of *x*, *t*). For various aspects of local and global existence, and uniqueness of solutions of (1.1), wave breaking phenomena, soliton-type solutions (‘peakons’), complete integrability aspects, such as infinitely many conservation laws, (bi-)Hamiltonian formalism, Bäcklund transformations, infinite dimensional symmetry groups, etc., we refer to the literature provided in Gesztesy & Holden (2002, 2003, ch. 5). The case of spatially periodic solutions, the corresponding inverse spectral problem, isospectral classes of solutions and quasi-periodicity of solutions with respect to time are discussed in Constantin (1997*a*,*b*, 1998*a*,*b*) and Constantin & McKean (1999). Moreover, algebro-geometric solutions of (1.1) and their properties are studied in Alber & Fedorov (2000, 2001), Alber *et al*. (2001) and Gesztesy & Holden (2002, 2003, ch. 5).

In §2, we recall the basic polynomial recursion formalism that defines the CH hierarchy using a zero-curvature approach. Section 3 recalls the stationary CH hierarchy and the associated algebro-geometric formalism. Section 4 provides a brief summary of self-adjoint canonical systems as needed in this paper, and §5 finally discusses the principal result of this paper, the class of real-valued, smooth and bounded algebro-geometric solutions of the CH hierarchy and the associated isospectral torus. We also briefly discuss real-valued algebro-geometric solutions with a cusp behaviour (cf. (5.80)).

We focus primarily on the case of stationary CH hierarchy solutions as the time-dependent case subordinates to the stationary one with respect to isospectral torus questions, a fact that is briefly commented on at the end of §5.

This paper should be viewed as a companion to our treatment (Gesztesy & Holden 2002, 2003, ch. 5) of the CH hierarchy and we refer to it for background material and pertinent references on the subject.

## 2. The CH hierarchy, recursion relations and hyperelliptic curves

In this section, we review the basic construction of the Camassa–Holm hierarchy using a zero-curvature approach following (Gesztesy & Holden 2002, 2003, ch. 5).

Throughout this section, we suppose the following hypothesis ().

In the stationary case, we assume that(2.1)In the time-dependent case (cf. (2.28)–(2.35)), we suppose(2.2)

We start by formulating the basic polynomial set-up. One defines recursively by(2.3)where is given by(2.4)One observes that is the resolvent of minus the one-dimensional Laplacian at energy parameter equal to −4, i.e.(2.5)The first coefficient reads as follows:(2.6)where *c*_{1} is an integration constant. Subsequent coefficients are non-local with respect to *u*. At each level, a new integration constant, denoted by *c*_{ℓ}, is introduced. Moreover, we introduce coefficients and by(2.7)Explicitly, one computes(2.8)Given hypothesis 2.1, one introduces the 2×2 matrix *U* by(2.9)and for each , the following 2×2 matrix *V*_{n} by(2.10)assuming *F*_{n}, *G*_{n} and *H*_{n} to be polynomials of degree *n* with respect to *z* and *C*^{∞} in *x*. Postulating the zero-curvature condition(2.11)one finds(2.12)(2.13)(2.14)From (2.12) to (2.14) one infers that(2.15)and hence(2.16)where the polynomial *R*_{2n+2} of degree 2*n*+2 is *x*-independent,(2.17)

Next, one makes the ansatz that *F*_{n}, *H*_{n} and *G*_{n} are polynomials of degree *n*, related to the coefficients , and by(2.18)Insertion of (2.18) into (2.12)–(2.14) then yields the recursion relations (2.3), (2.4) and (2.7) for and for *ℓ*=0, …, *n*. For fixed *n*∈, we obtain the recursion (2.7) for *h*_{ℓ} for *ℓ*=0, …, *n*−1 and(2.19)(When *n*=0 one directly gets .) Moreover, taking *z*=0 in (2.16) yields(2.20)In addition, one finds(2.21)Using the relations (2.19) and (2.7) permits one to write (2.21) as(2.22)Varying in (2.22) then defines the stationary CH hierarchy. We record the first few equations explicitly(2.23)

By definition, the set of solutions of (2.22), with *n* ranging in _{0}, represents the class of algebro-geometric CH solutions. If *u* satisfies one of the stationary CH equations in (2.22) for a particular value of *n*, then it satisfies infinitely many such equations of order higher than *n* for certain choices of integration constants . At times, it will be convenient to abbreviate (algebro-geometric) stationary CH solutions *u* simply as CH *potentials*.

Using equations (2.12)–(2.14), one can also derive individual differential equations for *F*_{n} and *H*_{n}. Focusing on *F*_{n} only, one obtains(2.24)and(2.25)Equation (2.25) leads to an explicit determination of the integration constants in the stationary CH equations (2.22) in terms of the zeros *E*_{0}=0, of the associated polynomial *R*_{2n+2} in (2.17). In fact, one can prove(2.26)where(2.27)

Next, we turn to the time-dependent CH hierarchy. Introducing a deformation parameter into *u* (i.e. replacing *u*(*x*) by ), the definitions (2.9), (2.10) and (2.18) of *U*, *V*_{n} and *F*_{n}, *G*_{n} and *H*_{n}, respectively, still apply. The corresponding zero-curvature relation then reads(2.28)which results in the following set of time-dependent equations:(2.29)(2.30)(2.31)Inserting the polynomial expressions for *F*_{n}, *H*_{n} and *G*_{n} into (2.30) and (2.31), respectively, first yields recursion relations (2.3) and (2.7) for *f*_{ℓ} and *g*_{ℓ} for . For fixed *n*∈, we obtain from (2.29) the recursion for *h*_{ℓ} for and(2.32)(When *n*=0 one directly gets .) In addition, one finds(2.33)Using relations (2.19) and (2.32) permits one to write (2.33) as(2.34)Varying *n*∈_{0} in (2.34) then defines the time-dependent CH hierarchy. We record the first few equations explicitly(2.35)Up to an inessential scaling of the (*x*, *t*_{1}) variables, with *c*_{1}=0 represents *the Camassa–Holm* equation as discussed in Camassa & Holm (1993).

## 3. The stationary algebro-geometric CH formalism

This section is devoted to a quick review of the stationary CH hierarchies and the corresponding algebro-geometric formalism as derived in Gesztesy & Holden (2002, 2003, ch. 5).

We start with the stationary hierarchy and suppose that *u* : ^{2}→ satisfies(3.1)and assume (2.3), (2.7), (2.9)–(2.11) and (2.16)–(2.22), keeping *n*∈_{0} fixed.

Recalling (2.17),(3.2)we introduce the (possibly singular) hyperelliptic curve _{n} of arithmetic genus *n* defined by(3.3)In the following, we will occasionally impose further constraints on the zeros *E*_{m} of *R*_{2n+2} introduced in (3.2) and assume that(3.4)We compactify _{n} by adding two points at infinity, *P*_{∞+}, *P*_{∞−}, with , still denoting its projective closure by _{n}. Hence, _{n} becomes a two-sheeted Riemann surface of arithmetic genus *n*. Points *P* on are denoted by *P*=(*z*, *y*), where denotes the meromorphic function on _{n} satisfying _{n}(*z*, *y*)=0.

For notational simplicity, we will usually tacitly assume that *n*∈ (the case *n*=0 being trivial).

In the following, the roots of the polynomials *F*_{n} and *H*_{n} will play a special role and, hence, we introduce on ×(3.5)temporarily assuming(3.6)Moreover, we introduce(3.7)(3.8)and(3.9)The branch of *y*(.) near is fixed according to(3.10)Due to assumption (3.1), *u* is smooth and bounded, and hence and share the same property. Thus, one concludes(3.11)taking multiplicities (and appropriate reordering) of the zeros of *F*_{n} and *H*_{n} into account.

Next, we introduce the fundamental meromorphic function on _{n} by(3.12)Assuming (3.4) and (3.6), the divisor of is given by(3.13)taking into account (3.10). Here we abbreviated(3.14)where , *m*∈, denotes the *m*th symmetric product of _{n}. If *h*_{0} is permitted to vanish at a point *x*_{1}∈, then for *x*=*x*_{1}, the polynomial is at most of degree *n*−1 (cf. (2.18)) and (3.13) is altered to(3.15)that is, one of the tends to as *x*→*x*_{1} (cf. also (3.36)). Analogously one can discuss the case of several approaching . Since this can be viewed as a limiting case of (3.13), we will henceforth not particularly distinguish the case *h*_{0}≠0 from the more general situation where *h*_{0} is permitted to vanish.

Given , one defines the associated Baker–Akhiezer vector on by(3.16)where(3.17)(3.18)Although *Ψ* is formally the analogue of the Baker–Akhiezer vector of the stationary CH hierarchy when compared to analogous definitions in the context of the KdV or AKNS hierarchies, its actual properties in a neighbourhood of its essential singularity feature characteristic differences to standard Baker–Akhiezer vectors as discussed in Gesztesy & Holden (2002, 2003, ch. 5).

The basic properties of *ϕ* and *Ψ* then read as follows.

*Suppose* (3.1), *assume the nth stationary CH equation* (2.22) *holds*, *and let* , . *Then ϕ satisfies the Riccati-type equation*(3.19)*as well as*(3.20)(3.21)(3.22)*while Ψ fulfils*(3.23)(3.24)(3.25)(3.26)(3.27)(3.28)(3.29)*In addition*, *as long as the zeros of* *are all simple for* , *an open interval*, , , *is meromorphic on* .

Next, we recall the Dubrovin-type equations for *μ*_{j}. Since in the remainder of this section, we will frequently assume _{n} to be non-singular, we list all restrictions on _{n} in this case,(3.30)

*Suppose* (3.1) *and the nth stationary CH equation* (2.22) *holds subject to the constraint* (3.30) *on an open interval* . *Moreover, suppose that the zeros μ*_{j}, *j*=1, …, *n*, *of F*_{n}(.) *remain distinct and non-zero on* . *Then* {}_{j=1, …, n} *defined by* (3.7), *satisfies the following first-order system of differential equations*:(3.31)*Next*, *assume* _{n} *to be non-singular and introduce the initial condition*(3.32)*for some* , *where* , *j*=1, …, *n*, *are assumed to be distinct*. *Then there exists an open interval* , *with* , *such that the initial value problem* (3.31) *and* (3.32) *has a unique solution* *satisfying*(3.33)*and μ*_{j}, *j*=1, …, *n*, *remain distinct and non-zero on Ω*_{μ}.

Combining the polynomial approach in §2 with (3.5) yields trace formulae for the CH invariants. For simplicity, we just record two simple cases.

*Suppose* (3.1), *assume the nth stationary CH equation* (2.22) *holds*, *and let* . *Then*(3.34)

(3.35)

Next, we turn to asymptotic properties of *ϕ* and *ψ*_{j}, *j*=1, 2.

*Suppose* (3.1), *assume the nth stationary CH equation* (2.22) *holds*, *and let* , *x*∈. *Then*(3.36)(3.37)*and*(3.38)

(3.39)

(3.40)

(3.41)

Since the representations of *ϕ* and *u* in terms of the Riemann theta function associated with _{n} (assuming _{n} to be non-singular) are not explicitly needed in this paper, we omit the corresponding details and refer to the detailed treatment in Gesztesy & Holden (2002, 2003, ch. 5) instead.

Finally, we will recall that solvability of the Dubrovin equations (3.31) on in fact implies equation (2.22) on *Ω*_{μ}.

*Fix n*∈, *assume* (3.30), *and suppose that* *satisfies the stationary Dubrovin equations* (3.31) *on an open interval* *such that μ*_{j}, *j*=1, …, *n*, *remain distinct and non-zero on Ω*_{μ}. *Then* *defined by*(3.42)*satisfies the nth stationary CH equation* (2.22), *i.e.*(3.43)

## 4. Basic facts on self-adjoint Hamiltonian systems

We now turn to the Weyl–Titchmarsh theory for singular Hamiltonian (canonical) systems and briefly recall the basic material needed in §5. This material is standard and can be found, for instance, in Kogan & Rofe-Beketov (1974), Hinton & Shaw (1981, 1983, 1984), Clark & Gesztesy (2002) and Lesch & Malamud (2003), and references therein.

(i) Define the 2×2 matrixand suppose , *j*,*k*=1,2, and , for a.e. *x*∈. We consider the Hamiltonian system(4.1)for a.e. *x*∈, where *z* plays the role of the spectral parameter, and where(4.2)Here denotes the set of locally absolutely continuous functions on and the *M*^{*} and denote the adjoint and transpose of a matrix *M*, respectively.

(ii) For all non-trivial solutions of (4.1), we assume the definiteness hypothesis (cf. Atkinson 1964, §9.1)(4.3)on every interval , *c*<*d*.

A simple example of a Hamiltonian system satisfying (4.3) is obtained whenfor some weight function , *w*>0 a.e. on , and a.e. on (cf. §5). Hypothesis 4.1 (ii) clearly holds in this case.

Next, we introduce the vector space (4.4)where denotes the standard scalar product in ^{2}. Fix a point *x*_{0}∈. Then the Hamiltonian system (4.1) is said to be in the *limit point case* at ∞ (respectively, −∞) if for some (and hence for all) , precisely one solution of (4.1) lies in (respectively, ). (By the analogue of Weyl's alternative, if (4.1) is not in the limit point case at , all solutions of (4.1) lie in for all . In the latter case, the Hamiltonian system (4.1) is said to be in the *limit circle case* at .)

To simplify matters for the remainder of this section, we will always suppose the limit point case at from now on.

Assume hypothesis 4.1 and suppose that the Hamiltonian system (4.1) is in the limit point case at .

An elementary example of a Hamiltonian system satisfying hypothesis 4.2 is given by the case where all entries of *A* and *B* are essentially bounded on (cf. §5).

When considering the Hamiltonian system (4.1) on the half-line (respectively, ), a self-adjoint (separated) boundary condition at the point *x*_{0} is of the type(4.5)where satisfies(4.6)In particular, the boundary condition (4.5) (with *α* satisfying (4.6)) is equivalent to with if and if . The special case will be of particular relevance in §5. Due to our limit point assumption at in hypothesis 4.2, no additional boundary condition at needs to be introduced when considering (4.1) on the half-lines and . The resulting full-line and half-line Hamiltonian systems are said to be self-adjoint on , and , respectively (assuming of course a boundary condition of the type (4.5) in the two half-line cases).

Next, we digress a bit and briefly turn to Herglotz functions and their representations in terms of measures, the focal point of Weyl–Titchmarsh theory (and hence spectral theory) of self-adjoint Hamiltonian systems.

Any analytic map is called a *Herglotz* function (here ). Similarly, any analytic map , , is called a *k*×*k* matrix-valued Herglotz function if for all .

Herglotz functions are characterized by a representation of the form(4.7)(4.8)(4.9)in the following sense: every Herglotz function admits a representation of the type (4.7) and (4.8) and, conversely, any function of the type (4.7) and (4.8) is a Herglotz function. Moreover, local singularities and zeros of *m* are necessarily located on the real axis and at most of first order in the sense that(4.10)(4.11)In particular, isolated poles of *m* are simple and located on the real axis, the corresponding residues being negative. Analogous results hold for matrix-valued Herglotz functions (Gesztesy & Tsekanovskii 2000 and references therein).

For subsequent purpose in §5, we also note that −1/*z* is a Herglotz function and compositions of Herglotz functions remain Herglotz functions. In addition, diagonal elements of a matrix-valued Herglotz function are Herglotz functions.

Returning to Hamiltonian systems on half-lines satisfying hypotheses 4.1 and 4.2, we now denote by the unique solution of (4.1) satisfying , , normalized by . Then the half-line Weyl–Titchmarsh function , associated with the Hamiltonian system (4.1) on and the fixed boundary condition at the point , is defined by(4.12)The actual normalization of was chosen for convenience only and is clearly immaterial in the definition of in (4.12).

One easily verifies that satisfies the following Riccati-type differential equation:(4.13)

Finally, the 2×2 Weyl–Titchmarsh matrix associated with the Hamiltonian system (4.1) on is then defined in terms of the half-line Weyl–Titchmarsh functions by(4.14)(4.15)One verifies that is a 2×2 matrix-valued Herglotz function. We emphasize that for any fixed *x*_{0}∈, *M*(*z*, *x*_{0}) contains all the spectral information of the self-adjoint Hamiltonian system (4.1) on (assuming hypotheses 4.1 and 4.2).

## 5. Real-valued algebro-geometric CH solutions and the associated isospectral torus

In our final and principal section, we study real-valued algebro-geometric solutions of the CH hierarchy associated with curves _{n} whose affine part is non-singular and determine the isospectral manifold of smooth and bounded CH solutions. We focus on the stationary case as this is the primary concern in this context and briefly comment on the time-dependent case at the end of this section.

To study the direct spectral problem, we first introduce the following assumptions.

Suppose(5.1)and let *u* be a real-valued solution of the *n*th stationary CH equation (2.22),(5.2)(i.e. *u* is a particular algebro-geometric CH potential), satisfying(5.3)

(5.4)

We start by noticing that the basic stationary equation (3.23),(5.5)is equivalent to the following Hamiltonian (canonical) system:(5.6)where(5.7)(5.8)In particular, due to assumptions (5.3) and (5.4), the Hamiltonian system (5.6) satisfies hypotheses 4.1 and 4.2. Explicitly, the Hamiltonian system (5.6) boils down to(5.9)(5.10)and upon eliminating results in a particular case of the weighted Sturm–Liouville problem(5.11)of the type(5.12)with ‘weight’ and constant coefficients *p*=*q*=1.

Introducing(5.13)we define(5.14)and(5.15)and analytically continue to . We also note the property(5.16)For notational convenience, we will occasionally call , , spectral gaps and the corresponding spectral gap endpoints.

Next, we introduce the cut plane(5.17)and the upper, respectively, lower sheets of _{n} by(5.18)with the associated charts(5.19)The two branches of the Baker–Akhiezer vector in (3.16) are then given by(5.20)and one infers from (3.38) that(5.21)Thus, introducing(5.22)and the two branches of on by(5.23)one infers from (4.12) and (5.21) that the Weyl–Titchmarsh functions associated with the self-adjoint Hamiltonian system (5.6) on the half-lines and the Dirichlet boundary condition indexed by at the point *x*∈ are given by(5.24)More precisely, (5.21) yields (5.24) only for sufficiently large . However, since by general principles are analytic in \, and by (3.12), are analytic in , one infers (5.24) by analytic continuation. In particular, (5.21) extends to all , i.e.(5.25)

Next, we mention a useful fact concerning a special class of Herglotz functions closely related to the problem at hand. The result must be well known to experts, but since we could not quickly locate a proof in the literature, we provide the simple contour integration argument below.

*Let P*_{N} *be a monic polynomial of degree N*. *Then* *is a Herglotz function if and only if one of the following alternatives applies*:

*N*=*n and*(5.26)*If**(5.26)**is satisfied*,*then**admits the Herglotz representation*(5.27)*and*(5.28)*If**(5.28)**is satisfied*,*then**admits the Herglotz representation*(5.29)

Since Herglotz functions are *O*(*z*) as and cannot vanish faster than *O*(1/*z*) as , we can confine ourselves to the range . We start with the case *N*=*n* and employ the following contour integration approach. Consider a closed oriented contour , which consists of the clockwise oriented semicircle centred at *E*_{0}, the straight line (oriented from left to right), the following part of the anticlockwise oriented circle of radius centred at *E*_{0}, , , and the straight line (oriented from right to left). Then, for *ϵ*>0 small enough and *R*>0 sufficiently large, one infers(5.30)Here we used (5.14) to compute the contributions of the contour integral along [*E*_{0}, *R*] in the limit *ϵ*↓0 and note that the integral over *C*_{R} tends to zero as *R*↑∞ since(5.31)Next, utilizing the fact that *P*_{n} is monic and using (5.14) again, one infers that represents a positive measure supported on *Σ* if and only if *P*_{n} has precisely one zero in each of the intervals , . In other words,(5.32)if and only if *P*_{n} has precisely one zero in each of the intervals , . The Herglotz representation (4.7) and (4.8) then finishes the proof of (5.27).

In the case where *N*=*n*+1, the proof of (5.28) follows along similar lines taking into account the additional residues at ±*i* inside , which are responsible for the real part on the right-hand side of (5.29).

Finally, in the case *N*=*n*+2, assume that is a Herglotz function. Then necessarily,(5.33)for some *a*∈, *b*≥0, and some finite (positive) measure *ω* supported on [*E*_{0}, 0], since(5.34)In particular, (5.33) implies(5.35)However, by (5.14), one immediately infers(5.36)This contradiction dispenses with the case *N*=*n*+2. ▪

Now we are in position to state the following result concerning the half-line and full-line Weyl–Titchmarsh functions associated with the self-adjoint Hamiltonian system (5.6). We denote by the Weyl–Titchmarsh *m*-functions corresponding to (5.6) associated with the half-lines and the Dirichlet boundary condition indexed by at the point *x*∈, and by the 2×2 Weyl–Titchmarsh matrix corresponding to (5.6) on (cf. (4.12), (4.14) and (4.15)). Moreover, denotes the open interior of and the real part of a matrix *M* is defined as usual by .

*Assume* *hypothesis* 5.1 *and let* , . *Then*(5.37)(5.38)*where* , , *is chosen such that*(5.39)*Moreover*,(5.40)(5.41)*where*(5.42)*The essential spectrum of the half-line Hamiltonian systems* (5.6) *on* (*with any self-adjoint boundary condition at x*) *as well as the essential spectrum of the Hamiltonian system* (5.6) *on* *is purely absolutely continuous and given by*(5.43)*The spectral multiplicities are simple in the half-line cases and of uniform multiplicity two in the full-line case*.

Equation (5.37) follows from (3.12), (5.14) and (5.24). Equation (5.40) is then a consequence of (3.20)–(3.22), (4.14), (4.15), (5.24) and (5.37). Different self-adjoint boundary conditions at the point *x* lead to different half-line Hamiltonian systems whose Weyl–Titchmarsh functions are related by a linear fractional transformation (cf. Clark & Gesztesy 2002) that leads to the invariance of the essential spectrum with respect to the boundary condition at *x*. In order to prove the Herglotz representation (5.38), one can follow the corresponding computation for Schrödinger operators with algebro-geometric potentials in Levitan (1987, §8.1). For this purpose, one first notes that by (5.29) also is a Herglotz function. A contour integration as in the proof of lemma 5.1 then proves(5.44)(5.45)The only difference compared to the corresponding argument in the proof of lemma 5.1 concerns additional (approximate) semicircles of radius *ϵ* centred at each *μ*_{j}(*x*), , in the upper and lower complex half-planes. Whenever , the limit *ϵ*↓0 picks up a residue contribution displayed in the sum over *j* in (5.44). This contribution vanishes, however, if . In this case, by (4.10) and by (2.17). Equation (5.45) then follows from (3.7) and the sign of *ϵ*_{j}(*x*) must be chosen according to (5.39) in order to guarantee non-positive residues in (5.45) (cf. (4.10)).

Next, we apply the Lagrange interpolation formula. If *Q*_{n−1} is a polynomial of degree *n*−1, then(5.46)Since *F*_{n} and *G*_{n} are monic polynomials of degree *n*, we can apply (5.46) to *Q*_{n−1}=*G*_{n}−*F*_{n} and obtain(5.47)Insertion of (5.47) into (5.37) then yields(5.48)and hence (5.38) follows by inserting (5.45) into (5.48). Equations (5.41) and (5.42) are clear from the matrix analogue of (4.9).

The statement (5.43) for the essential half-line spectra then follows from the fact that the measure in the Herglotz representation (5.38) of (as a function of *z*) is supported on the set in (5.13), with a strictly positive density on the open interior of . The transformation then yields (5.43) and since half-line spectra with a regular endpoint *x* have always simple spectra, this completes the proof of our half-line spectral assertions. The full-line case follows in exactly the same manner since the corresponding 2×2 matrix-valued measure *Ω* in the Herglotz representation (5.41) of (as a function of *z*) also has support and rank equal to 2 on . ▪

Returning to direct spectral theory, we note that the two spectral problems (5.6) on associated with the vanishing of the first and second component of at *x*, respectively, are clearly self-adjoint since they correspond to the choices and in (4.5). Hence, a comparison with (3.5), (3.26) and (3.27) necessarily yields . Thus, we will assume the convenient eigenvalue ordering(5.49)The zeros of belong to the Dirichlet spectral problem associated with the Hamiltonian system (5.12) (respectively, the weighted Sturm–Liouville problem (5.12)) on . A comparison with (3.26) then relates the zeros , , of in (3.5) to the Dirichlet spectrum of (5.6) (respectively, (5.12)) on . The correspondence between each *μ*_{j} and the related spectral point of the Dirichlet problem (5.6) (respectively, (5.12)) on is of course effected by the transformation . In contrast to this, the zeros of do not belong to the Neumann spectrum associated with the Hamiltonian system (5.6) (respectively, the weighted Sturm–Liouville problem (5.12)) on . In fact, by (5.9), zeros of correspond to a mixed boundary condition of the type . By (3.27), this relates the zeros , , of in (3.5) to the spectrum of (5.6) (respectively, (5.12)) on corresponding to the self-adjoint boundary condition .

Combining lemma 5.1 with the Herglotz property of the 2×2 Weyl–Titchmarsh matrix then yields the following refinement of theorem 3.2.

*Assume* *hypothesis* 5.1. *Then* , *with the projections* , , *the zeros of* in (3.5), *satisfies the first-order system of differential equations* (3.31) *on Ω*_{μ}= *and*(5.50)*Moreover*,(5.51)*In particular*, *changes sheets whenever it hits E*_{2j−1} *or E*_{2j} *and its projection μ*_{j}(*x*) *remains trapped in* *for all* *and x*∈. *The analogous statements apply to* *and one infers*(5.52)

Since is a 2×2 Herglotz matrix, its diagonal elements are Herglotz functions. Thus,(5.53)are Herglotz functions (the left-hand sides with respect to , the right-hand sides with respect to *z*) and the interlacing properties (5.51) and (5.52) then follow from (5.28) and (5.26). ▪

Combining the interlacing property (5.51) with (2.18), (2.19) and (2.20) yields (cf. also (3.35))(5.54)in accordance with (5.4). Moreover, since by (5.52) the *v*_{j}(*x*) also remain trapped in the intervals for all *x*∈, none of the can reach and hence on (cf. the discussion surrounding (3.15)). Actually,(5.55)since is a Herglotz function (cf. (5.27)).

The zeros , of , which are related to eigenvalues of the Hamiltonian system (5.6) on associated with the boundary condition , in fact, are related to left and right half-line eigenvalues of the corresponding Hamiltonian system restricted to the half-lines and , respectively. Indeed, by (5.22) and (5.25), depending on whether or , *μ*_{j}(*x*) is related to a left or right half-line eigenvalue associated with the Dirichlet boundary condition . A careful investigation of the sign of the right-hand sides of the Dubrovin equations (3.30) (combining (5.1), (5.14) and (5.18)), then proves that the *μ*_{j}(*x*) related to right (respectively, left) half-line eigenvalues of the Hamiltonian system (5.6) associated with the boundary condition , are strictly monotone increasing (respectively, decreasing) with respect to *x*, as long as the *μ*_{j} stay away from the right (respectively, left) endpoints of the corresponding spectral gaps . Here we purposely avoided the limiting case where some of the *μ*_{k}(*x*) hit the boundary of the spectral gaps, , since the half-line eigenvalue interpretation is lost as there is no eigenfunction satisfying in this case. In fact, whenever an eigenvalue *μ*_{k}(*x*) hits a spectral gap endpoint, the associated point on _{n} crosses over from one sheet to the other (equivalently, the corresponding left half-line eigenvalue becomes a right half-line eigenvalue and vice versa) and, accordingly, strictly increasing half-line eigenvalues become strictly decreasing half-line eigenvalues and vice versa. In particular, using the appropriate local coordinate (respectively, ) near *E*_{2k} (respectively, *E*_{2k−1}), one verifies that *μ*_{k}(*x*) does not pause at the endpoints *E*_{2k} and *E*_{2k−1}.

Next, we turn to the inverse spectral problem and determine the isospectral manifold of real-valued, smooth and bounded CH solutions.

Our basic assumptions then will be the following.

Suppose(5.56)fix *x*_{0}∈ and assume that the initial data(5.57)for the Dubrovin equations (3.31) are constrained by(5.58)

*Assume* *hypothesis* 5.2. *Then the Dubrovin initial value problem* *(3.31), (5.57)* *and* *(5.58)* *has a unique solution* *satisfying*(5.59)*and the projections μ*_{j} *remain trapped in the intervals* , , *for all x*∈,(5.60)*Moreover*, *u defined by the trace formula* (3.34), *i.e.*(5.61)*satisfies* *hypothesis* 5.1, *i.e.*(5.62)(5.63)(5.64)*and the nth stationary CH equation*(5.65)*with integration constants c*_{ℓ} *in* (5.65) *given by* , , *according to* (2.26) *and* (2.27).

Given initial data constrained by , , one concludes from the Dubrovin equations (3.31) and the sign properties of on the intervals , , described in (5.14), that the solution *μ*_{j}(*x*) remains in the interval as long as stays away from the branch points . In case hits such a branch point, one can use the local chart around (*E*_{m},0), with local coordinate , , , to verify (5.59) and (5.60). Relations (5.61)–(5.63) are then evident from (5.59), (5.60) and(5.66)In the course of the proof of theorem 3.1 presented in Gesztesy & Holden (2002, 2003, §5.3), one constructs the polynomial formalism (*F*_{n}, *G*_{n}, *H*_{n}, *R*_{2n+2}, etc.) and then obtains identity (3.35) as an elementary consequence. The latter immediately proves (5.64). Finally, (5.65) follows from theorem 3.1 (with *Ω*_{μ}=).

*Fix* *and assume the ordering* (5.56). *Then the isospectral manifold of smooth and bounded real-valued solutions* *of* *is given by a real n-dimensional torus* ^{n}.

The discussion in remark 5.2 and theorem 5.3 shows that the motion of each on _{n} proceeds topologically on a circle and is uniquely determined by the initial data , . More precisely, the initial data(5.67)are topologically equivalent to data of the type(5.68)the sign of depending on . If some of the , then the determination of the sheet and hence the sign in (5.68) becomes superfluous and is eliminated from (5.68). Indeed, since by (2.16),(5.69) is determined up to a sign unless hits a spectral gap endpoint in which case and the sign ambiguity disappears. The *n* data in (5.68) (properly interpreted if ) can be identified with circles. Since the latter are independent of each other, the isospectral manifold of real-valued, smooth and bounded solutions of is given by a real *n*-dimensional torus ^{n}.

For simplicity, we only focused on the case The opposite case is completely analogous and results in a reflection of

*E*_{m}, , and , , about*z*=0, etc.The time-dependent case also offers nothing new. Higher-order CH

_{r}flows drive each around the same circles as in the stationary case in complete analogy to the familiar KdV case.

In summary, one observes that the reality problem for smooth and bounded solutions of the CH hierarchy, assuming the ordering (5.56) (respectively, the one obtained upon reflection with respect to *z*=0), parallels that of the KdV hierarchy with the basic self-adjoint Lax operator (the one-dimensional Schrödinger operator) replaced by the self-adjoint Hamiltonian system (5.6).

The following result was found in response to a query of Igor Krichever who inquired about the significance of the eigenvalue ordering (5.56) (or the one obtained upon reflection at *z*=0). As it turns out, such an ordering is crucial if one is interested in smooth algebro-geometric solutions on .

*Suppose*(5.70)*Fix x*_{0}∈ *and assume that the initial data*(5.71)*for the Dubrovin equations* (3.31) *are constrained by*(5.72)*Then there exists a set Ω*_{μ}⊂ *of the type*(5.73)*such that the Dubrovin initial value problem* (3.31), (5.57) *and* (5.58) *has a unique solution* *satisfying*(5.74)*and the projections μ*_{j} *remain trapped in the intervals* , , *for all x*∈*Ω*_{μ},(5.75)*Moreover*, *u defined by the trace formula* (3.34), *i.e.*(5.76)*satisfies*(5.77)(5.78)*and the nth stationary CH equation*(5.79)*with integration constants c*_{ℓ} *in* (5.65) *given by* , , *according to* (2.26) *and* (2.27). *At each ξ*_{k}, *u*_{x} *exhibits a singularity of the type*(5.80)*In particular*, , . *The isospectral manifold corresponding to* (5.70)–(5.72) *is then given by* ^{n−1}×.

One can follow the proof of theorem 5.3 and corollary 5.1 with one important twist, though, since the right-hand side of the Dubrovin equation of blows up as . For notational convenience and without loss of generality, we may assume *E*_{1}=0 (and hence *j*_{0}=1) in the following. Recalling the Dubrovin equations (3.31), one verifies that its solutions are smooth with respect to *x* as long as *μ*_{1} stays away from *E*_{1}=0. Varying *x*∈, the sign restrictions on in terms of the right-hand side of the corresponding equation in (3.31) eventually accelerate *μ*_{1} into *E*_{1}=0 as *x* tends to some *ξ*_{k}∈, and we now analyse what happens to all *μ*_{j} for *x* in a neighbourhood of *ξ*_{k}. Recalling the local coordinate , , near *E*_{1}=0 and hence introducing(5.81)and the corresponding point , the Dubrovin equation for *μ*_{j} becomes for *x* near *ξ*_{k},(5.82)(5.83)for some constant *C*_{1}≠0. (Here we implicitly assume that no other *μ*_{j}, simultaneously hits *E*_{2j−1} or *E*_{2j} as *x*→*ξ*_{k}. Otherwise one simply resorts to the proper local coordinate for such a *μ*_{j}. We omit the details.) To treat the singularity of as for *x*→*ξ*_{k}, we now resort to a well-known trick described, for instance, in Hille (1976, theorem 3.2.2) in the context of scalar first-order differential equations. Instead of looking for solutions , *μ*_{j} as functions of *x*, we now look for , as functions of , where we denote , . Then (5.82) and (5.83) turn into(5.84)(5.85)Since the right-hand sides in (5.84) and (5.85) are holomorphic with respect to the *n* variables for near zero and near , , equations (5.84) and (5.85) yield solutions holomorphic with respect to near . In particular, since(5.86)for some constants , , one obtains(5.87)(5.88)Thus, inverting , one observes(5.89)(5.90)and hence,(5.91)and(5.92)The singular behaviour (5.91) and (5.92) repeats itself after each revolution of *μ*_{1} around its circle and occurs whenever *μ*_{1} passes again through *E*_{1}=0, giving rise to the exceptional set in (5.73). Hence, , , while blows up whenever *x* approaches an element of . The rest of the discussion follows as in theorem 5.3 and corollary 5.1. Since is not an admissible initial condition in (5.72), one point must be removed from the circle associated to *μ*_{1}, which topologically results in instead of *S*^{1} and hence in the non-compact isospectral manifold ^{n−1}×. ▪

Thus, smooth algebro-geometric CH solutions require *E*_{0}=0 or *E*_{2n+1}=0.

Finally, we briefly turn to the time-dependent case.

Suppose that satisfies(5.93)

The basic problem in the analysis of algebro-geometric solutions of the CH hierarchy consists in solving the time-dependent *r*th CH flow with initial data a stationary solution of the *n*th equation in the hierarchy. More precisely, given *n*∈_{0}, consider a solution of the *n*th stationary CH equation associated with _{n} and a given set of integration constants . Next, let *r*∈_{0}; we intend to construct a solution *u* of the *r*th CH flow CH_{r}(*u*)=0 with for some . To emphasize that the integration constants in the definitions of the stationary and the time-dependent CH equations are independent of each other, we indicate this by adding a tilde on all the time-dependent quantities. Hence, we shall employ the notation , , , , , , , , etc. in order to distinguish them from *V*_{n}, *F*_{n}, *G*_{n}, *H*_{n}, *f*_{ℓ}, *g*_{ℓ}, *h*_{ℓ}, *c*_{ℓ}, etc. in the following. In addition, we will follow a more elaborate notation inspired by Hirota's *τ*-function approach and indicate the individual *r*th CH flow by a separate time variable *t*_{r}∈.

Summing up, we are seeking a solution *u* of(5.94)(5.95)for some , , where *u* satisfies (5.93).

We pause for a moment to reflect on the pair of equations (5.94) and (5.95): as it turns out (cf. Gesztesy & Holden 2002, 2003, §5.4), it represents a dynamical system on the set of algebro-geometric solutions isospectral to the initial value *u*^{(0)}. By isospectral we here allude to the fact that for any fixed , the solution of (5.94) and (5.95) is a stationary solution of (5.95),(5.96)associated with the fixed underlying algebraic curve _{n}. Put differently, is an isospectral deformation of *u*^{(0)} with *t*_{r} the corresponding deformation parameter. In particular, traces out a curve in the set of algebro-geometric solutions isospectral to *u*^{(0)}.

Thus, relying on this isospectral property of the CH flows, we will go a step further and assume (5.95) not only at but also for all . Hence, we start with(5.97)(5.98)where (cf. (2.18))(5.99)and(5.100)(5.101)(5.102)(5.103)(5.104)(5.105)(5.106)(5.107)for fixed . Here , , , , and for , *s*=0, …, *r*, are defined as in (2.3) and (2.7) with *u*(*x*) replaced by *u*(*x*, *t*_{r}), etc. and with appropriate integration constants. Explicitly, (5.97) and (5.98) are equivalent to(5.108)(5.109)(5.110)and(5.111)

(5.112)

(5.113)

One observes that equations (2.3)–(2.25) apply to *F*_{n}, *G*_{n}, *H*_{n}, *f*_{ℓ}, *g*_{ℓ} and *h*_{ℓ}, and (2.3)–(2.8) and (2.18), with *n* replaced by *r* and *c*_{ℓ} replaced by , apply to , , , , and . In particular, the fundamental identity (2.16),(5.114)holds as in the stationary context and the hyperelliptic curve _{n} is still given by(5.115)Here we are still assuming (3.4), i.e.(5.116)

The independence of (5.114) of *t*_{r}∈ can be interpreted as follows. The *r*th KdV flow represents an isospectral deformation of the curve _{n} in (5.115), in particular, the branch points of _{n} remain invariant under these flows(5.117)Together with the comments following (5.95), this shows that isospectral torus questions are conveniently reduced to the study of the stationary hierarchy of CH flows since time-dependent solutions just trace out a curve in the isospectral torus defined by the stationary hierarchy. This is of course in complete agreement with other completely integrable 1+1-dimensional hierarchies such as the KdV, Toda and AKNS hierarchies.

## Acknowledgments

We are indebted to Mark Alber, Roberto Camassa, Adrian Constantin, Yuri Fedorov and Igor Krichever for stimulating discussions. This research was supported in part by the Research Council of Norway.

## Footnotes

One contribution of 15 to a Theme Issue ‘30 years of finite-gap integration’.

- © 2007 The Royal Society