The toda shock problem

Venakides, Stephanos; Deift, Percy; Oba, Roger M.

doi:10.1002/cpa.3160440823

Cited by 76 publications

(77 citation statements)

References 9 publications

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“…The single commutation method for Jacobi operators first appeared in Gesztesy et al [117]. A special case of the double commutation method can be found in [235]. The general case of the double commutation method seems due to [110] which is one of the the main sources for this chapter.…”

Section: Notes On Literaturementioning

confidence: 94%

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: 94%

Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

324

349

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“…By the transfer theorem, the capacity of Σ a is √ a. As is shown in [35], a Jacobi matrix J of period 2 and whose spectrum is exactly Σ a is, for instance, given by …”

Section: Examplementioning

confidence: 97%

Capacities and Jacobi Matrices

Sebbar

Falliero

2003

Proceedings of the Edinburgh Mathematical Society

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In this paper, we use the theorem of Burchnall and Shaundy to give the capacity of the spectrum σ(A) of a periodic tridiagonal and symmetric matrix. A special family of Chebyshev polynomials of σ(A) is also given. In addition, the inverse problem is considered: given a finite union E of closed intervals, we study the conditions for a Jacobi matrix A to exist satisfying σ(A) = E. We relate this question to Carathéodory theorems on conformal mappings.

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“…The continuous analogue of this phenomenon has been well studied in the context of the Korteweg-de Vries equation [42,44,79] . There have also been detailed studies of dispersive numerical schemes [27,33,45,60,74,76,80] . In this section we shall discuss three canonical dispersive schemes for the Hopf equation, and speculate variations of their model equations.…”

Section: Dispersive Schemesmentioning

confidence: 99%

On discreteness of the Hopf equation

Liu

2008

Acta Math. Appl. Sin. Engl. Ser.

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The principle aim of this essay is to illustrate how different phenomena is captured by different discretizations of the Hopf equation and general hyperbolic conservation laws. This includes dispersive schemes, shock capturing schemes as well as schemes for computing multi-valued solutions of the underlying equation. We introduce some model equations which describe the behavior of the discrete equation more accurate than the original equation. These model equations can either be conveniently discretized for producing novel numerical schemes or further analyzed to enrich the theory of nonlinear partial differential equations.

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The toda shock problem

Cited by 76 publications

References 9 publications

Jacobi Operators and Completely Integrable Nonlinear Lattices

Jacobi Operators and Completely Integrable Nonlinear Lattices

Capacities and Jacobi Matrices

On discreteness of the Hopf equation

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