Longtime asymptotics of the periodic Toda lattice under shortrange perturbations
Abstract
We compute the longtime asymptotics of periodic (and slightly more generally of algebrogeometric finitegap) solutions of the doubly infinite Toda lattice under a shortrange perturbation. In particular, we prove that the perturbed lattice asymptotically approaches a modulated lattice. More precisely, let g be the genus of the hyperelliptic curve associated with the unperturbed solution. We show that, apart from the phenomenon of solitons travelling in a quasiperiodic background, the n/tpane contains g + 2 areas where the perturbed solution is close to a finitegap solution on the same isospectral torus. In between there are g + 1 regions where the perturbed solution is asymptotically close to a modulated lattice which undergoes a continuous phase transition (in the Jacobian variety) and which interpolates between these isospectral solutions. In the special case of the free lattice (g = 0), the isospectral torus consists of just one point and we recover the known result. Both the solutions in the isospectral torus and the phase transition are explicitly characterized in terms of Abelian integrals on the underlying hyperelliptic curve. Our method relies on the equivalence of the inverse spectral problem to a vector RiemannHilbert problem defined on the hyperelliptic curve and generalizes the socalled nonlinear stationary phase/steepest descent method for RiemannHilbert problem deformations to Riemann surfaces.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 July 2012
 DOI:
 10.1063/1.4731768
 arXiv:
 arXiv:0705.0346
 Bibcode:
 2012JMP....53g3706K
 Keywords:

 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 Mathematical Physics
 EPrint:
 38 pages, 1 figure. This version combines both the original version and arXiv:0805.3847