## Abstract

Recently, the string density problem, considered in the pioneering work of M. G. Krein, has arisen naturally in connection with the Camassa–Holm equation for shallow water waves. In this paper we review the forward and inverse string density problems, with some numerical examples, and relate it to the Camassa–Holm equation, with special reference to multi-peakon/anti-peakon solutions. Under stronger assumptions, the Camassa–Holm spectral problem and the string density problem can be transformed to the Schrödinger spectral problem and its inverse problem. Recent results exploiting this transformation are reviewed briefly.

## 1. Introduction

In a series of papers in the 1950s, Krein investigated the direct and inverse problems associated with a string of variable density (Krein 1951, 1952*a*,*b*, 1953, 1954; see also Dym & McKean 1976; Dym & Kravitsky 1978*a*,*b*). Krein viewed these problems as a natural outgrowth of topics ranging from the work of Stieltjes on continued fractions to the moment problem and spectral representation of Hermitian operators (see Krein (1952*a*,*b*) and appendix 3 in Gorbachuk & Gorbachuk (1997)).

Recently, the string density problem has come up in connection with the Camassa–Holm (CH) equation. The CH equation was first identified as a ‘completely integrable’ equation by Fuchssteiner (1981) and Fuchsteiner & Fokas (1981). It was derived from the Green–Naghdi equations for shallow water waves by Camassa & Holm (1993). Johnson (2002) rederived the CH equation using classical approximation methods.

Camassa and Holm identified a class of special peaked solutions (peakons) that interact in ways similar to the Korteweg–de Vries (KdV) solitons. As is well known, KdV is associated with a linear spectral problem for a one-dimensional Schrödinger operator, and explicit formulae for the special (soliton and multi-soliton) solutions can be obtained from the solutions of the Schrödinger inverse problem due to Gel'fand & Levitan (1951) and Marčenko (1955). The spectral problem associated with the CH equation has an energy-dependent potential and the (general) inverse theory is more difficult to exploit. The spectral problem on the line can be transformed to the string density problem on a finite interval and in the case of the special (multi-peakon/anti-peakon) solutions, the inverse problem becomes one that had been solved by Stieltjes (1894).

In §2 of this brief review paper we outline the direct and inverse theory for a general string. Some experimental results for the inverse problem are exhibited in §3. Section 4 outlines the connection with the CH equation and the peakon/anti-peakon solutions. The connection with the Liouville transformation and factorization methods is reviewed in §5, which concludes with discussion of the recent literature.

## 2. The direct and inverse string density problems

The linearized equation for vibrations of a string of length 1 with density *m* isIt is natural to impose Dirichlet boundary conditions (fixed endpoints)Under these conditions, the energy is independent of time *t*. The standing wave solutions with initial condition ∂*u*(*x*, 0)/∂*t*=0 have the formwhere *u*_{ν} is a generalized eigenfunction for the associated ODE(2.1)with . The energy is the *L*^{2} norm .

We allow the ‘density’ *m* to be a measure *m*=d*M*, where *M* is a non-decreasing function on the interval *I*=[0, 1]. We assume that *M* is *normalized*: left continuous with 0=*M*(0)<*M*(1−)=*M*(1)<∞. The ODE (2.1) for a function *φ* with initial conditions *φ*(0, *λ*)=0 and *Dφ*(0, *λ*)=1 can be interpreted as integral equations for *φ* and *Dφ*,(2.2)

(2.3)There are corresponding equations for the function *ψ* normalized at *x*=1,

The generalized spectrum, i.e. the set of zeros of *φ*(1, *λ*), consists of a finite or countable set {*λ*_{ν}}. We setThen, *φ*_{ν}=−*Dφ*_{ν}(1)*ψ*_{ν}. By *spectral data* for the density d*M* we mean the set of generalized eigenvalues {*λ*_{ν}} together with either of the sets of constants {*a*_{ν}} or {*b*_{ν}}, where

The following are the principal results on the forward and inverse spectral problems in this generality.

*The generalized spectrum of a density dM is positive and simple*,*The associated constants* {*a*_{ν}} *and* {*b*_{ν}} *are related by*(2.4)*Moreover*,(2.5)

There are three main ingredients. One is the identity(2.6)where *f*=*D*^{2}*u*+*λmu*. This specializes to a collection of identities that imply that the roots of *φ*(1,*λ*) are positive and simple, and that the derivatives *Dφ*_{ν} are orthogonal in *L*^{2}(*I*, d*x*).

The second ingredient is an analysis of the iterative solutions of the integral equations (2.2) and (2.3),Estimates on the *u*_{n} and *v*_{n} imply that *φ*(1, .) and *Dφ*(1, *λ*) are entire of order 1/2 and, consequently, have product representations(2.7)

The third ingredient is the *Weyl function*(2.8)This has a product representation from equation (2.7) and an additive representation(2.9)as the Stieltjes transform of the measure .

Replacing *φ* by *ψ* in this construction leads to a second Weyl function(2.10)The residues are *a*_{0}=*b*_{0}=1 and, for *ν*>0,and they are related by equation (2.4).

Combining information from these three ingredients leads to the identities(2.11)which give equation (2.5).

▪We say that the density *m*=d*M* is *discrete* if *M* is piecewise constant, i.e.(2.12)Such a density can be calculated algebraically from its spectral data (see Beals *et al*. 1999, 2000 and the discussion below). This leads to an inversion method.

*Let* {*λ*_{ν}}, {*a*_{ν}} *be the spectral data of a density* d*M*. *For each positive integer n*, *there is a unique discrete measure m*_{n}=d*M*_{n}, *with spectral data* *and* . *The associated functions M*_{n} *converge to M in L*^{p}, 1≤*p*<∞.

This proof, like the previous one, is related to the analysis of the Weyl function. Suppose that d*M* is the discrete measure (2.12). Let *l*_{j}=*x*_{j}−*x*_{j−1}. The function *φ*(., *λ*) is continuous and piecewise linear. The density d*M* is encoded by the data {*l*_{j}}, {*m*_{j}}. Conversely, the spectral data {*λ*_{ν}}, {*a*_{ν}} are encoded in the discrete measure *ρ*, which determines its Stieltjes transform *W* and which conversely is determined by *W*. The Weyl function has a (unique) continued fraction expansion(2.13)The result of Stieltjes (1894) expresses the coefficients of the continued fraction (2.13) in terms of ratios of products of minors of the moment matrix (*A*_{jk}), where

Although we assumed that this spectral data (or the measure *ρ*) was obtained from a given discrete density d*M*, the Stieltjes inversion process can be applied to any formal datato show that it is spectral data for a (unique) discrete density d*M*.

Suppose now that {*λ*_{ν}}, {*a*_{ν}} are spectral data, then they and their related constants {*b*_{ν}} satisfyIn view of the previous remarks, we may assume that these sets are infinite and look for a corresponding density d*M*. We note at the start that uniqueness can be deduced from uniqueness results for the Neumann problem for the ‘dual string’ (Kac & Krein 1974; Dym & McKean 1976).

For each positive *n*, there is a unique discrete density d*M*_{n}, with {*λ*_{1}, …, *λ*_{n}} and {*a*_{1}, …, *a*_{n}} as spectral data. Owing to the assumptions and the identities (2.5), the positive discrete measures d*M*_{n} are uniformly bounded. Therefore, there are subsequences that are weak^{*} convergent. The corresponding functions are uniformly bounded and almost everywhere convergent, hence convergent in *L*^{p} norm, 1≤*p*<∞. It follows that the integral operators in equations (2.2) and (2.3) corresponding to the densities *M*_{n} from the subsequence converge in Hilbert–Schmidt norm and the corresponding functions {*Dφ*^{(n)} (., *λ*)} converge in *L*^{2} uniformly with respect to *λ* in a bounded set. The same is true for the {*Dψ*^{(n)}}, so for any fixed index *ν*, . The entire functions *φ*^{(n)}(1, *λ*) converge uniformly in bounded sets, so the zeros converge. Thus, the spectral data for the density *M* that is the weak^{*} limit is the given data {*λ*_{ν}}, {*a*_{ν}}. By uniqueness, this weak^{*} limit is independent of the particular subsequence. Therefore, the original sequence converges.

Most treatments of the string problem emphasize the Neumann problem. We have emphasized the Dirichlet problem because it is more closely connected to the CH equation (see §4).

It is well known that under strong enough regularity assumptions on

*M*, the Liouville transform takes the string problem to the spectral problem for a one-dimensional Schrödinger equation (see the discussion in §5). Again, it is the connection with the CH equation that prompts us to work with minimal regularity. Indeed, the multi-peakon solution of the CH equation corresponds exactly to the case of a discrete density, i.e. the case in which*M*is a step function.For the general CH equation, one would like to solve the string problem with a possibly negative density:

*M*of bounded variation rather than monotone. This brings one even farther from the range of applicability of the Liouville transform. In the (signed) discrete case, the Stieltjes method still works for actual spectral data. However, formal data may fail to be actual data (but see §3).It has been known since the paper of Borg (1946) that one spectrum is not, in general, enough to determine uniquely a Sturm–Liouville potential, but that two spectra are. The situation is similar in the string problem. We have worked here interchangeably with the data {

*λ*_{ν}}, {*a*_{ν}} and the Weyl function*W*. The latter is uniquely determined by its zeros {*μ*_{ν}} and its poles {*λ*_{ν}}. The latter are the Dirichlet spectrum, while the former, as the zeros of*Dφ*(1,*λ*), are the spectrum for the Dirichlet/Neumann boundary conditions*u*(0)=0,*D**u*(1)=0. The uniqueness result implies that this pair of spectra determines implicitly all other spectra, but the relation can be made (somewhat) more explicit. For simplicity, we consider only the Neumann spectrum. Let*Χ*be the solution of equation (2.1), with boundary conditions*Χ*(1,*λ*)=1,*DΧ*(1,*λ*)=0. Then,Differentiating with respect to*x*and evaluating at*x*=0 shows that the Neumann spectrum is the set of*λ*, such that*D*_{ψ}(0,*λ*)*D*_{ψ}(1,*λ*)=−1, orBarcilon (1982) proposed the spectral approximation method of theorem 2.2 as a computation method for the beam spectral problem.

## 3. Numerical examples

In this section we illustrate theorem 2.2 for a few simple examples of strings with piecewise constant *M*(*x*), using the first *n*=40 eigenvalues and residues {*a*_{ν}} to construct a discrete string with masses *m*_{j} at points partitioning the interval into subintervals of lengths *l*_{j}. We graphas a function of *x*_{j}=*l*_{n+1}+*l*_{n}+⋯+*l*_{n+2−j}, which represents the mass distribution relative to the right end of the string. (Note that this reconstruction procedure has a bias towards the right end of the string, where it is more accurate. One could use and reconstruct from the left end instead.) Note that the third example violates the assumption that *M* be non-decreasing and the spectrum is infinite in both the directions. One can nevertheless attempt the analogous discrete approximation.

*Constant positive density* (figure 1). We choose *m*(*x*)=1. The eigenvalues and eigenfunctions are *λ*_{ν}=(*νπ*)^{2}, *φ*_{ν}(*x*)=sin *νπx*.

*Piecewise constant density* (figure 2). We take *m*(*x*)=4/9, 0≤*x*≤3/4 and *m*(*x*)=4, 3/4<*x*≤1.

The corresponding mass distribution (with *x* denoting the distance to the right end of the string) isThe total mass *M*(1)=4/3. The eigenvalues are *λ*_{ν}=(*νπ*)^{2}, with eigenfunctions

*Piecewise constant density of both signs* (figure 3). The density is taken to be *m*(*x*)=1, 0≤*x*≤1/2 and *m*(*x*)=−1, 1/2<*x*≤1. The eigenfunctions arewhere satisfies tan(*ω*)+tanh(*ω*)=0. Eigenvalues are symmetric with respect to 0 and we take a truncation of the spectrum that is symmetric about *λ*=0.

## 4. The Camassa–Holm equation

The Camassa–Holm equation can be written compactly as a system(4.1)It is the compatibility condition for the overdetermined system(4.2)

The first of these equations (for a function independent of *t*) is the associated linear spectral problem. Under an assumption about decay of *m* at , it will have solutionsFor a discrete set of values *λ*_{ν}, these solutions will be multiples of each other,and under the CH flow, the values {*λ*_{ν}} are fixed, while it follows from the asymptotics of equation (4.2) that the connection coefficients evolve according to(4.3)

The peakon/anti-peakon solutions of equation (4.2) have the formcorresponding to the discrete measurePositive *m*_{j} give ‘peakons’, which move to the right, negative *m*_{j} give ‘anti-peakons’, which move to the left. The Camassa–Holm evolution in this case is a 2*n*-dimensional Hamiltonian system with canonical variables {*x*_{j}} and {*m*_{j}} and Hamiltonian (see Camassa & Holm 1993)

This linear spectral problem is transformed to a (formal) string problem on the interval [−1, 1] by the transformation (Beals *et al*. 1998)Then, the first equation in equation (4.2) becomes the string equation with density *m*^{*},Under the decay assumption above, *m*^{*} is bounded at the endpoints. The functions *φ* and *ψ* transform to functions *φ*^{*} and *ψ*^{*} corresponding to *m*^{*}. If *m*^{*} does not change sign, then the analysis of §2 applies. Under the Camassa–Holm flow, the corresponding residues {*a*_{ν}} or {*b*_{ν}} evolve in the same way as the connection coefficients {*c*_{ν}} (4.3). Therefore, the initial-value problem can be solved by solving the inverse string density problem and transforming back to *m* on the line (Beals *et al*. 1999, 2000).

If *m* changes sign, there are difficulties with any approach to equation (4.1). For the case of a discrete measure, the Stieltjes inversion method described in §2 is purely algebraic, so it continues to produce explicit formulae for and thus for *m*_{j}(*t*). At certain times and locations, some minors in denominators will vanish; this corresponds to peakon/anti-peakon ‘collisions’, at which the function *u* remains continuous, although one or more of the *m*_{j} blow up (Beals *et al*. 2000). This shows that, on the one hand, formal scattering data for a (signed) discrete string may not be actual data, but, on the other hand, this is not fatal for the CH flow. For more details on the Stieltjes method, as well as the relation between d*M* and orthogonal polynomials, see Beals *et al*. (2000).

The generalized spectral problem for the discrete string can be mapped to an ordinary spectral problem for a Jacobi matrix, by an algebraic analogue of the Liouville transformation (which does not require positivity). When the CH flow is transformed in this way, it fits into a natural commuting family of Jacobi flows that includes the Toda flow as studied by Moser (1975) (see Beals *et al*. 2001).

## 5. Factorization and inverse problems

Solution of an inverse scattering problem typically reduces to solving a factorization problem, of either the Riemann–Hilbert type or the operator type. This feature is well known in the case of the Schrödinger equation in connection with the KdV equation. The inverse problem can either be cast as a Riemann–Hilbert problem for the jump of a suitably normalized wave function across the real axis of the complex spectral plane or as the solution of the Gel'fand &Levitan (1951) or Marčenko (1955) equations.

Gel'fand and Levitan were motivated by Krein's attempt to extend the theory of the classical moment problem, and it is interesting to consider their approach to the inverse spectral problem in this light. The solution of the inverse problem arising in the multi-peakon solutions of the Camassa–Holm equation can be formulated in the following way (Beals *et al*. 2000). Given a positive discrete measure , construct a set of orthonormal polynomials with respect to d*μ*, i.e. . Such a sequence of polynomials is constructed by orthogonalizing the set with respect to d*μ*(*λ*) using the Gram–Schmidt procedure. As is well known, this is equivalent to factoring a matrix *X*=*QR*, where *Q* is an orthogonal matrix and *R* is an upper triangular matrix. The so-called *QR* factorization plays a fundamental role in many problems involving integrable systems (Symes 1980, 1981).

Gel'fand and Levitan considered the operator *L*_{y}=−*y*″+*q* on (0, ∞) with boundary conditions *y*(0)=1,*y*′(0)=*h*. By spectral theory, the wave functions *φ*(*x*, *λ*) satisfy *Lφ*=*λφ* and the boundary conditions satisfy the completeness relation(5.1)Gel'fand and Levitan solved the inverse problem as follows. Given a suitable positive measure d*ρ*(*λ*) on the line, construct a family of functions *φ*(*x*, *λ*) that satisfy (5.1) by ‘orthogonalizing’ the functions with respect to d*ρ*. The solutions of *Lφ*=*λφ* can be represented in the formfor some kernel function *K*(*x*, *t*); such a Volterra integral operator is the continuous analogue of a triangular matrix. Assuming that the solutions of the orthogonalization problem have such a representation, Gel'fand and Levitan obtained the integral equationwhere

The potential *q* and parameter *h* are obtained from the kernel *K*. This approach works also for Sturm–Liouville problems on finite intervals (0, *a*) and is easily extended to include Dirichlet boundary conditions (by replacing with ).

When *m*>0 is sufficiently regular, the Liouville transformation can be used to convert the CH spectral operator(5.2)to a Schrödinger operator *D*^{2}+*λ*−*q*. In fact, the spectral problemis transformed via the Liouville transformationto the Schrödinger spectral problemwhere and the potential *q* isThis transformation has been employed by Constantin (2001), Lenells (2002, 2004), Constantin & Lenells (2003*a*,*b*), Johnson (2003) and McKean (2003). The Liouville transformation also works in the case of the periodic Camassa–Holm equation (see Constantin 1998; Constantin & McKean 1999; Korotyaev 2004). McKean considers potentials *m*>0 for which *a*_{±}=±∞, whereas Constantin considers a modified Camassa–Holm equation, namelyfor which the spectral operator is *D*^{2}−*z*(*m*+*ω*)−(1/4). For *m*>0, the Schrödinger operator then has continuous spectrum in the interval (−∞,−1/4*ω*]. WhenConstantin (2001) has shown that there are at most finitely many bound states contained in the interval (−1/4*ω*, 0). An inversion of the nonlinear relation between *q* and *m* has been given by Constantin & Lenells (2003*a*,*b*). Using the Liouville transformation, McKean has shown the equivalence of the Camassa–Holm hierarchy and the KdV flows at negative order under these assumptions. This transformation has also been used by McKean in the periodic case (McKean 2003) to transform the spectral problem to Hill's equation. Johnson uses the inverse transformation to pull back the reflectionless Schrödinger potential with two or three eigenvalues, which yield the two- and three-soliton KdV solutions, to corresponding special solutions of the CH equation.

For , the interval for the Schrödinger operator is finite, but, in general, the Schrödinger potential *q*(*y*) blows up at the endpoints, so the resulting Sturm–Liouville problem is singular. On the other hand, for positive *m* in the Schwartz class, Li (2005) has given a very interesting direct solution to the initial-value problem for the CH equation, using a formulation of the CH equation in characteristic coordinates due to Camassa (2003). Li formulates the problem as an initial-value problem for a Hilbert–Schmidt operator *K*(*t*) that satisfies the equation of Lax typewhere *B*(*K*) is the skew part of *K*, in perfect analogy with the Toda lattice and the *QR* factorization. As a consequence, one obtains global existence of solutions to the CH initial-value problem.

As noted in §4, equation (5.2) can be transformed to a string problem on a finite interval. This transformation is independent of *m*. When *m* is positive and sufficiently regular, one can proceed to use the Liouville transformation to convert this string problem to a Schrödinger spectral problem (the composition of these two transformations will be the Liouville transformation just discussed).

## Acknowledgments

The research of J.S. is supported by the Natural Sciences and Engineering Research Council of Canada.

## Footnotes

One contribution of 13 to a Theme Issue ‘Water waves’.

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