We provide an overview of elliptic algebro-geometric solutions of the KdV and AKNS hierarchies, with special emphasis on Floquet theoretic and spectral theoretic methods. Our treatment includes an effective characterization of all stationary elliptic KdV and AKNS solutions based on a theory developed by Hermite and Picard.
The theory of commuting linear differential expressions has received a lot of attention since Lax presented his description of the KdV hierarchy by Lax pairs (P, L). Gesztesy and the present author have established a relationship of this circle of ideas with the property that all solutions of the differential equations Ly = zy, z ∈ C, are meromorphic. In… (More)
We study the connections between Gelfand–Dickey (GD) systems and their modified counterparts , the Drinfeld–Sokolov (DS) systems in the case of general matrix–valued coefficients with entries in a commutative algebra over an arbitrary field. Our main results describe auto–Bäcklund transformations for the GD hierarchy based on Miura–type transformations… (More)
If A is a ω-periodic matrix Floquet's theorem states that the differential equation y = Ay has a fundamental matrix P (x) exp(Jx) where J is constant and P is ω-periodic, i.e., P (x) = P * (e 2πix/ω). We prove here that P * is rational if A is bounded at the ends of the period strip and if all solutions of y = Ay are meromorphic. This version of Floquet's… (More)
In this paper two classical theorems by Levinson and Marchenko for the inverse problem of the Schrödinger equation on a compact interval are extended to finite trees. Specifically, (1) the Dirichlet eigenvalues and the Neumann data of the eigenfunctions determine the potential uniquely (a Levinson-type result) and (2) the Dirichlet eigenvalues and a set of… (More)
If the differential expressions P and L are polynomials (over C) of another differential expression they will obviously commute. To have a P which does not arise in this way but satisfies [P, L] = 0 is rare. Yet the question of when it happens has received a lot of attention since Lax presented his description of the KdV hierarchy by Lax pairs (P, L). In… (More)
Stationary solutions of higher order KdV equations play an important role for the study of the KdV equation itself. They give rise to the coefficients of the associated Lax pair (P, L) for which P and L have an algebraic relationship (and are therefore called algebro-geometric). This paper gives a sufficient condition for rational and simply periodic… (More)
We consider the class of CMV operators with super-exponentially decaying Verblunsky coefficients. For these we define the concept of a resonance. Then we prove the existence of Jost solutions and a uniqueness theorem for the inverse resonance problem: Given the location of all resonances, taking multiplicities into account, the Verblunsky coefficients are… (More)