From Toda to KdV

Abstract

For periodic Toda chains with a large number $\mathit{N}$ of particles we consider states which are $\mathit{N}^{-2}$-close to the equilibrium and constructed by discretizing arbitrary given C$^2$−functions with mesh size $\mathit{N}^{-1}$. 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 $\mathit{N} \rightarrow \infty$. To this end we construct two Hill operators $\mathit{H}\pm$ Show more