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 language | English (US) |
|---|---|
| Pages (from-to) | 305-333 |
| Number of pages | 29 |
| Journal | Communications in Mathematical Physics |
| Volume | 221 |
| Issue number | 2 |
| DOIs | |
| State | Published - Jul 2001 |
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ASJC Scopus subject areas
- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
Cite this
Long time behavior of the continuum limit of the Toda lattice, and the generation of infinitely many gaps from C∞ initial data. / Kuijlaars, A. B J; Mclaughlin, Kenneth D T.
In: Communications in Mathematical Physics, Vol. 221, No. 2, 07.2001, p. 305-333.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Long time behavior of the continuum limit of the Toda lattice, and the generation of infinitely many gaps from C∞ initial data
AU - Kuijlaars, A. B J
AU - Mclaughlin, Kenneth D T
PY - 2001/7
Y1 - 2001/7
N2 - 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.
AB - 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.
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U2 - 10.1007/s002200100482
DO - 10.1007/s002200100482
M3 - Article
VL - 221
SP - 305
EP - 333
JO - Communications in Mathematical Physics
JF - Communications in Mathematical Physics
SN - 0010-3616
IS - 2
ER -