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From Toda to KdV
Analysis of PDEs · Submitted 20 Sep 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim To this end we construct two Hill operators $H\_\pm$ -- such operators come up in the Lax pair formulation of the Korteweg-de Vries equation -- and prove by methods of semiclassical analysis that the asymptotics as $N \rightarrow \infty $ of the eigenvalues at the edges of the spectrum of $L\_N$ are of the form $\pm (2-(2N)^{-2} \lambda ^\pm \_n + \cdots )$ where $(\lambda ^\pm \_n)\_{n \geq 0}$ are the eigenvalues of $H\_\pm $.
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The abstract, sentence by sentence
For periodic Toda chains with a large number $N$ of particles we consider states which are $N^{-2}$-close to the equilibrium and constructed by discretizing arbitrary given $C^2-$functions with mesh size $N^{-1}.$ Our aim is to describe the spectrum of the Jacobi matrices $L\_N$ appearing in the Lax pair formulation of the dynamics of these states as $N \to \infty$. To this end we construct two Hill operators $H\_\pm$ -- such operators come up in the Lax pair formulation of the Korteweg-de Vries equation -- and prove by methods of semiclassical analysis that the asymptotics as $N \rightarrow \infty $ of the eigenvalues at the edges of the spectrum of $L\_N$ are of the form $\pm (2-(2N)^{-2} \lambda ^\pm \_n + \cdots )$ where $(\lambda ^\pm \_n)\_{n \geq 0}$ are the eigenvalues of $H\_\pm $. In the bulk of the spectrum, the eigenvalues are $o(N^{-2})$-close to the ones of the equilibrium matrix. As an application we obtain asymptotics of a similar type of the discriminant, associated to $L\_N$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | For periodic Toda chains with a large number $N$ of particles we consider states which are $N^{-2}$-close to the equilibrium and constructed by discretizing arbitrary given $C^2-$functions with mesh size $N^{-1}.$ Our aim is to describe the spectrum of the Jacobi matrices $L\_N$ appearing in the Lax pair formulation of the dynamics of these states as $N \to \infty$. | background 0.61 | 0.71 | 0.04 | asserted 0.69 |
| s1 | To this end we construct two Hill operators $H\_\pm$ -- such operators come up in the Lax pair formulation of the Korteweg-de Vries equation -- and prove by methods of semiclassical analysis that the asymptotics as $N \rightarrow \infty $ of the eigenvalues at the edges of the spectrum of $L\_N$ are of the form $\pm (2-(2N)^{-2} \lambda ^\pm \_n + \cdots )$ where $(\lambda ^\pm \_n)\_{n \geq 0}$ are the eigenvalues of $H\_\pm $. | result 0.54 | 0.95 | 0.68 | asserted 1.00 |
| s2 | In the bulk of the spectrum, the eigenvalues are $o(N^{-2})$-close to the ones of the equilibrium matrix. | result 1.00 | 0.90 | 0.12 | asserted 1.00 |
| s3 | As an application we obtain asymptotics of a similar type of the discriminant, associated to $L\_N$. | result 0.87 | 0.93 | 0.05 | asserted 1.00 |
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The paper
Original abstract
For periodic Toda chains with a large number $N$ of particles we consider states which are $N^{-2}$-close to the equilibrium and constructed by discretizing arbitrary given $C^2-$functions with mesh size $N^{-1}.$ Our aim is to describe the spectrum of the Jacobi matrices $L\_N$ appearing in the Lax pair formulation of the dynamics of these states as $N \to \infty$. To this end we construct two Hill operators $H\_\pm$ -- such operators come up in the Lax pair formulation of the Korteweg-de Vries equation -- and prove by methods of semiclassical analysis that the asymptotics as $N \rightarrow \infty $ of the eigenvalues at the edges of the spectrum of $L\_N$ are of the form $\pm (2-(2N)^{-2} \lambda ^\pm \_n + \cdots )$ where $(\lambda ^\pm \_n)\_{n \geq 0}$ are the eigenvalues of $H\_\pm $. In the bulk of the spectrum, the eigenvalues are $o(N^{-2})$-close to the ones of the equilibrium matrix. As an application we obtain asymptotics of a similar type of the discriminant, associated to $L\_N$.
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3 arXiv papers cite this in 13 sentences
Papers citing it · 8 sentences without a confident label
Compares with it 1 paper
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(In the papers [ref] , the size $R$ can be arbitrarily large.)
Birkhoff coordinates for the Toda Lattice in the limit of infinitely many particles with an application to FPU · Birkhoff coordinates for the Toda lattice · 0.98
Cites it as background 3 papers
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First results in this project were obtained in the papers [ref] .
Birkhoff coordinates for the Toda Lattice in the limit of infinitely many particles with an application to FPU · Birkhoff coordinates for the Toda lattice · 0.82
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Results on the behaviour of the integrable structure of Toda for large $N$ have been recently obtained in a series of papers [ref] .
Some analytic results on the FPU paradox · Toda lattice · 0.95
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It was shown in [ref] that the KdV equation can be viewed as a continuum approximation to the Toda lattice in the limit of long wavelengths and small amplitude [ref] , and that both systems are derived as limiting cases of a generalized equation, see also the recent work [ref] .
Numerical study of the long wavelength limit of the Toda lattice · Introduction · 0.80
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