Non-Linear Equations of Korteweg-De Vries Type, Finite-Zone Linear Operators, and Abelian Varieties

Dubrovin, Boris; Matveev, V. I.; Novikov, S. P.

doi:10.1070/rm1976v031n01abeh001446

Cited by 609 publications

(561 citation statements)

References 34 publications

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“…Gesztesy and Holden are also working on a monograph on hierarchies of soliton equations and their algebro-geometric solutions [103]. We refer, in particular, to the extensive treatments in [25], [74], [75], [76], [168], [177], and [184]. In the special context of the Toda equations we refer to [3], [75], [76], [156], [159], [168], [170], and [175].…”

Section: Notes On Literaturementioning

confidence: 99%

Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

324

349

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Abstract. This book is intended to serve both as an introduction and a reference to spectral and inverse spectral theory of Jacobi operators (i.e., second order symmetric difference operators) and applications of these theories to the Toda and Kac-van Moerbeke hierarchy.Starting from second order difference equations we move on to self-adjoint operators and develop discrete Weyl-Titchmarsh-Kodaira theory, covering all classical aspects like Weyl m-functions, spectral functions, the moment problem, inverse spectral theory, and uniqueness results. Next, we investigate some more advanced topics like locating the essential, absolutely continuous, and discrete spectrum, subordinacy, oscillation theory, trace formulas, random operators, almost periodic operators, (quasi-)periodic operators, scattering theory, and spectral deformations.Then, the Lax approach is used to introduce the Toda hierarchy and its modified counterpart, the Kac-van Moerbeke hierarchy. Uniqueness and existence theorems for the initial value problem, solutions in terms of Riemann theta functions, the inverse scattering transform, Bäcklund transformations, and soliton solutions are discussed.Corrections and complements to this book are available from:

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Section: Notes On Literaturementioning

confidence: 99%

Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

324

349

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“…The converse statement has not been proved so far. The direct statement is proved similarly to the fundamental theorem of [1] for l = 1 by the scheme presented in a more general, suitable and invariant form in the survey [2] (Chap. II).…”

Section: Theoremmentioning

confidence: 98%

“…Then, the operator L 4 , defined by (1), (2), has nonsingular real periodic coefficients and is a semibounded self-adjoint operator in L 2 (R).…”

Section: Theoremmentioning

confidence: 99%

Spectral theory of commuting operators of rank two with periodic coefficients

Novikov¹,

Grinevich²

1982

Funct Anal Its Appl

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In the Novikov-Krichever formula for a fourth-order operator L 4 , occurring in a commuting pair of pank two and genus one, there is an arithmetical error; the operator L 4 has the form(one of the terms d/dx a has been omitted), whereFor all the quotations and formulas see the paper by Grinevich in this issue.Theorem 1. The operator L 4 is formally symmetric if and only if a = 0, i.e., λ 1 = λ 2 (the constant γ 0 = 0 modulo the semiperiods of the function ℘).In this case the formulas for the coefficients are strongly simplified:(2)Assume that the following conditions hold : 1) the Riemann surface Γ is real (i.e., g 1 , g 2 are real ); 2) the function λ(x) is real ; 3) at the points where λ = 0 we have λ = 0, λ 2 = P 3 (λ). Then, the operator L 4 , defined by (1), (2), has nonsingular real periodic coefficients and is a semibounded self-adjoint operator in L 2 (R).For any linear ordinary differential operator L (with respect to x) with periodic coefficients there is defined a "monodromy matrix"T (λ), i.e., a shift operator on the period of solutions of the equation Lψ = λψ, written in a certain basis. The order of the matrixT (λ) is equal to the order k of the operator L. The eigenvectors ψ q (x, λ) of the operatorT (λ) are called the Bloch eigenfunctions (or the Floquet functions). The function ψ q (x, λ) is meromorphic in λ on the Riemann surfaceΓ over the λ-plane of k sheets, whose points are the pairs Q q = (λ, q), q = 1, . . . , k.Theorem 3. Assume that the operator L of order k = nl occurs in the commuting pair [A, L] = 0 of rank l with Riemann surface Γ of finite genus {P (A, L) = 0}. Then, the monodromy matrixT (λ), λ ∈ C, written in the basis of the BakerAkhiezer eigenfunctions ϕ α (x, P i ), α = 1, . . . , l, defines a matrixT * (P) of order

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“…Dirichlet spectra and the moment problem were first combined in the pioneering work by Kac and van Moerbeke [23], [24], [29]. The Jacobi inversion problem in connection with Dirichlet divisors appeared in Date and Tanaka [12] (see also [36]) and simultaneously in Dubrovin, Matveev, and Novikov [15] with further developments in [28], [30], [31]. (The algebro-geometric method is presented in great detail in [6].)…”

Section: Introductionmentioning

confidence: 99%

On isospectral sets of Jacobi operators

1996

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Abstract. We consider the inverse spectral problem for a class of reflectionless bounded Jacobi operators with empty singularly continuous spectra. Our spectral hypotheses admit countably many accumulation points in the set of eigenvalues as well as in the set of boundary points of intervals of absolutely continuous spectrum. The corresponding isospectral set of Jacobi operators is explicitly determined in terms of Dirichlet-type data.

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Non-Linear Equations of Korteweg-De Vries Type, Finite-Zone Linear Operators, and Abelian Varieties

Cited by 609 publications

References 34 publications

Jacobi Operators and Completely Integrable Nonlinear Lattices

Jacobi Operators and Completely Integrable Nonlinear Lattices

Spectral theory of commuting operators of rank two with periodic coefficients

On isospectral sets of Jacobi operators

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