For periodic Toda chains with a large number N of particles we consider states which areN(-2)-close to the equilibrium and constructed by discretizing arbitrary given C-2 -functions with mesh sizeN(-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 -> infinity. 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 -> infinity of the eigenvalues at the edges of the spectrum of L-N are of the form +/-(2 - (2N)(-2) lambda(+/-)(n) + ...) where (lambda(+/-)(n))(n >= 0) 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.
From Toda to KdV / D. Bambusi, T. Kappeler, T. Paul. - In: NONLINEARITY. - ISSN 0951-7715. - 28:7(2015), pp. 2461-2496.
From Toda to KdV
D. BambusiPrimo
;
2015
Abstract
For periodic Toda chains with a large number N of particles we consider states which areN(-2)-close to the equilibrium and constructed by discretizing arbitrary given C-2 -functions with mesh sizeN(-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 -> infinity. 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 -> infinity of the eigenvalues at the edges of the spectrum of L-N are of the form +/-(2 - (2N)(-2) lambda(+/-)(n) + ...) where (lambda(+/-)(n))(n >= 0) 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.File | Dimensione | Formato | |
---|---|---|---|
pdf.pdf
accesso riservato
Tipologia:
Publisher's version/PDF
Dimensione
435.54 kB
Formato
Adobe PDF
|
435.54 kB | Adobe PDF | Visualizza/Apri Richiedi una copia |
Pubblicazioni consigliate
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.