Peakons, strings, and the finite Toda lattice

Abstract

As is well-known, the Toda lattice flow may be realized as an isospectral flow of a Jacobi matrix. A bijective map from a discrete string problem with positive weights to Jacobi matrices allows the pure peakon flow of the Camassa-Holm equation to be realized as an isospectral Jacobi flow as well. This gives a unified picture of the Toda, Jacobi, and multipeakon flows, and leads to explicit solutions of the Jacobi flows via Stieltjes' determination of the continued fraction expansion of a Stieltjes transform. A simple modification produces a bijection from generalized strings, with positive and negative weights, to singular Jacobi matrices, and thus brings peakon/antipeakon flows into the same picture.

Strings, Waves, Drums: Spectra and Inverse Problems

“…In analogy with the situations discussed above, we expect additional information such as {a ν }, a ν = ||Dϕ ν || −2 to be necessary, in addition to {λ ν }, in order to determine the measure m uniquely. The fact that the data {λ ν }, {a ν } determines the measure m uniquely is a consequence of the work of Kreȋn; see also [12], [13]. The theory lies much deeper than the corresponding theory for the one-dimensional Schrödinger operator sketched above.…”

“…The Euler-Poincaré characteristic of an n-dimensional manifold M is Poincaré's generalization of (12.11): 13) where b k is the number of k-dimensional simplexes in a decomposition of M ; it can also be taken to be the k-th Betti number of the simplicial complex, which is itself a topological invariant. It is a consequence of Poincaré duality that χ(M ) = 0 if the dimension n is odd (a result that we return to in Chapter 14).…”

An application of Pfaffians to multipeakons of the Novikov equation and the finite Toda lattice of BKP type

“…This was proved by Constantin and Strauss [16,17]. (3) There exists a bijection between a certain Krein type string problem, which in turn is equivalent to the CH peakon problem, and the Jacobi spectral problem, the isospectral flow of which gives the celebrated Toda lattice [6]. In a well defined sense, the CH peakon ODEs could be regarded as a negative flow of the finite Toda lattice.…”

Algebro-Geometric Solutions of the Camassa–Holm hierarchy

“…Soliton-type solutions (called "peakons") were extensively studied due to their unusual non-meromorphic (peak-type) behavior, which features a discontinuity in the x-derivative of u with existing left and right derivatives of opposite sign at the peak. In this context we refer, for instance, to [3], [5], [7], [8], [10], [12], [13], [14], [17], [18], [51]. Integrability aspects such as infinitely many conservation laws, (bi-)Hamiltonian formalism, Bäcklund transformations, infinite dimensional symmetry groups, etc., are discussed, for instance, in [17], [18], [38], [41] (see also [42]), [57].…”

“…That the equations define a smooth vector field was first observed by Shkoller in the case of periodic [58] and Dirichlet [59] boundary conditions, which directly leads to the corresponding local existence theory. Scattering data and their evolution under the CH flow are determined in [11] and intimate relations with the classical moment problem and the finite Toda lattice are worked out in [12], [13], and [14]. 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 [20], [21], [22], and [29].…”