Wave phenomena of the Toda lattice with steplike initial data

TL;DR

This paper surveys the long-time asymptotic behavior of the Toda lattice with steplike initial data, employing nonlinear steepest descent analysis and providing explicit formulas for shock and rarefaction solutions.

Contribution

It introduces an extension of the nonlinear steepest descent method with a specific $g$-function and derives explicit formulas for asymptotic solutions including overlapping spectra cases.

Findings

Explicit formulas for Toda shock and rarefaction asymptotics

Numerical simulations confirming theoretical results

Analysis of overlapping background spectra cases

Abstract

We give a survey of the long-time asymptotics for the Toda lattice with steplike constant initial data using the nonlinear steepest descent analysis and its extension based on a suitably chosen $g$-function. Analytic formulas for the leading term of the asymptotic solutions of the Toda shock and rarefaction problems (including the case of overlapping background spectra) are given and complemented by numerical simulations. We provide an explicit formula for the modulated solution in terms of Abelian integrals on the underlying hyperelliptic Riemann surface.