Canonically Conjugate Variables for the Korteweg-de Vries Equation and the Toda Lattice with Periodic Boundary Conditions

Flaschka, H.; McLaughlin, David W.

doi:10.1143/ptp.55.438

Cited by 245 publications

(175 citation statements)

References 5 publications

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“…Since the situation is parallel to the case of the periodic Toda chains [8], we first review that case, then turn to the case of periodic dressing chains.…”

Section: Deriving Hamiltonian Systemmentioning

confidence: 99%

Spectral Curve, Darboux Coordinates and Hamiltonian Structure of Periodic Dressing Chains

Takasaki

2003

Commun. Math. Phys.

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A chain of one-dimensional Schrödinger operators connected by successive Darboux transformations is called the "Darboux chain" or "dressing chain". The periodic dressing chain with period N has a control parameter α. If α = 0, the N -periodic dressing chain may be thought of as a generalization of the fourth or fifth (depending on the parity of N ) Painlevé equations . The N -periodic dressing chain has two different Lax representations due to Adler and to Noumi and Yamada. Adler's 2 × 2 Lax pair can be used to construct a transition matrix around the periodic lattice. One can thereby define an associated "spectral curve" and a set of Darboux coordinates called "spectral Darboux coordinates". The equations of motion of the dressing chain can be converted to a Hamiltonian system in these Darboux coordinates. The symplectic structure of this Hamiltonian formalism turns out to be consistent with a Poisson structure previously studied by Veselov, Shabat, Noumi and Yamada.

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“…Since the situation is parallel to the case of the periodic Toda chains [8], we first review that case, then turn to the case of periodic dressing chains.…”

Section: Deriving Hamiltonian Systemmentioning

confidence: 99%

Spectral Curve, Darboux Coordinates and Hamiltonian Structure of Periodic Dressing Chains

Takasaki

2003

Commun. Math. Phys.

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“…During the period 1974-1976, many important aspects of the finite-gap integration method for the discrete Toda-like models were worked out (Kac & van Moerbeke 1975;Date & Tanaka 1976;Dubrovin et al 1976;Flaschka & McLaughlin 1976). It was clear from the spectral theory of the Jacobi matrices that for the discrete models the finite-gap solutions contain all solutions periodic with respect to the discrete (lattice) space variable.…”

Section: (I ) Matrix Differential Operators Zero-curvature Representmentioning

confidence: 99%

30 years of finite-gap integration theory

Matveev

2007

Phil. Trans. R. Soc. A.

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The method of finite-gap integration was created to solve the periodic KdV initial problem. Its development during last 30 years, combining the spectral theory of differential and difference operators with periodic coefficients, the algebraic geometry of compact Riemann surfaces and their Jacobians, the Riemann theta functions and inverse problems, had a strong impact on the evolution of modern mathematics and theoretical physics. This article explains some of the principal historical points in the creation of this method during the period 1973-1976, and briefly comments on its evolution during the last 30 years.

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“…Motivated by the last applications of the action variables the present author, Veselov and Dubrovin investigated them and constructed a theory of "algebrogeometric" finite dimensional P.B. for the systems, integrable by the method of Riemann surfaces (see [27,28]; these ideas were initiated in some calculations of the papers [29,30]). …”

Section: Weakly Deformed Soliton Lattices and Their Hamiltonian Hydromentioning

confidence: 99%

Differential Geometry and Hydrodynamics of Soliton Lattices

Novikov

1993

Springer Series in Nonlinear Dynamics

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Canonically Conjugate Variables for the Korteweg-de Vries Equation and the Toda Lattice with Periodic Boundary Conditions

Cited by 245 publications

References 5 publications

Spectral Curve, Darboux Coordinates and Hamiltonian Structure of Periodic Dressing Chains

Spectral Curve, Darboux Coordinates and Hamiltonian Structure of Periodic Dressing Chains

30 years of finite-gap integration theory

Differential Geometry and Hydrodynamics of Soliton Lattices

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