Long time behavior of the continuum limit of the Toda lattice, and the generation of infinitely many gaps from C initial data

A. B J Kuijlaars, Kenneth D T Mclaughlin

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7 Citations (Scopus)

Abstract

We analyze a continuum limit of the finite non-periodic Toda lattice through an associated constrained maximization problem over spectral density functions. The maximization problem was derived by Deift and McLaughlin using the Lax-Levermore approach, initially developed for the zero dispersion limit of the KdV equation. It encodes the evolution of the continuum limit for all times, including evolution through shocks. The formation of gaps in the support of the maximizer is indicative of oscillations in the Toda lattice and the lack of strong convergence of the continuum limit. For large times, the maximizer tends to have zero gaps, which is the continuum analogue of the sorting property of the finite lattice. Using methods from logarithmic potential theory, we show that this behavior depends crucially on the initial data. We exhibit initial data for which the zero gap ansatz holds uniformly in the spatial parameter (at large times), and other initial data for which this uniformity fails at all times. We then construct an example of C smooth initial data generating, at a later time, infinitely many gaps in the support of the maximizer, while for even larger times, all gaps have closed.

Original languageEnglish (US)
Pages (from-to)305-333
Number of pages29
JournalCommunications in Mathematical Physics
Volume221
Issue number2
DOIs
StatePublished - Jul 2001

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Toda Lattice
Continuum Limit
Long-time Behavior
continuums
Zero
Logarithmic Potential
Spectral Density Function
potential theory
KdV Equation
Potential Theory
classifying
Strong Convergence
Sorting
Uniformity
Shock
Continuum
shock
Tend
Oscillation
analogs

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics

Cite this

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