Asymptotics for Supersonic Soliton Propagation in the Toda Lattice Equation By L. A. Cisneros and A. A. Minzoni We study the problem of the adjustment of an initial condition to an exact supersonic soliton solution of the Toda latice equation. Also, we study the problem of soliton propagation in the Toda lattice with slowly varying mass impurities. In both cases we obtain the full numerical solution of the soliton evolution and we develop a modulation theory based on the averaged Lagrangian of the discrete Toda equation. Unlike previous problems with coherent subsonic solutions we need to modify the averaged Lagrangian to obtain the coupling between the supersonic soliton and the subsonic linear radiation. We show how this modified modulation theory explains qualitatively in simple terms the evolution of a supersonic soliton in the presence of impurities. The quantitative agreement between the modulation solution and the numerical result is good. 1. Introduction The Toda lattice represents an anharmonic one-dimensional lattice system with the nearest neighbor interaction of the exponential type, therefore it covers a wide range of interaction potentials from the harmonic oscillator to the strong nonlinear one. This discrete system is known to be completely integrable through the inverse scattering method [1]. The soliton solution to the equation Address for correspondence: L. A. Cisneros, Graduate Program in Mathematical Sciences, Facultad de Ciencias, Universidad Nacional Aut´ onoma de M´ exico, Apdo. 20-726, 01000 M´ exico, D.F., Mexico; e-mail: cisneros@math.unm.edu STUDIES IN APPLIED MATHEMATICS 120:333–349 333 C  2008 by the Massachusetts Institute of Technology Published by Blackwell Publishing, 350 Main Street, Malden, MA 02148, USA, and 9600 Garsington Road, Oxford, OX4 2DQ, UK.