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

We analyze a continuum limit of the finite non-periodic Toda lattice throug h an associated constrained maximization problem over spectral density func tions. The maximization problem was derived by Deift and McLaughlin using t he Lax-Levermore approach, initially developed for the zero dispersion limi t 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 init ial data. We exhibit initial data for which the zero gap ansatz holds unifo rmly 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 -infinity 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.