From Toda to KdV. (22nd June 2015)

• Record Type: Journal Article
• Authors: Bambusi, D; Kappeler, T; Paul, T
• Is Part Of: Nonlinearity. Volume 28:Number 7(2015:Jul.)
• Page Start: 2461
• Page End: 2496
• Publication Date: 2015-06-22
• Subjects: toda lattice -- KdV equation -- semiclassical limit; 37K10 -- 70F45 -- 81Q20; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/0951-7715/28/7/2461
• Languages: English
• ISSNs: 0951-7715

Abstract

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 this end we construct two Hill operators H ± —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 → ∞ of the eigenvalues at the edges of the spectrum of L N are of the form where are the eigenvalues of H ± . 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 .