Shock waves in the Toda lattice: Analysis

Known analytical results are used to analyze molecular-dynamics experiments of shock waves in the one-dimensional Toda lattice. (This lattice provides a physically realistic model which contains the hard-sphere and harmonic lattices as limits.) Both explicit solutions and rather general theoretical properties have been employed. The leading edge of the shock front is well represented quantitatively by a single isolated solition. Once compression is properly taken into account, the interior of the shock wave is accurately described by a slowly varying Toda wave train. A sharp transition in the dynamical response exists as the shock strength passes a critical value; this critical value is identified mathematically by the spectral transform for the Toda lattice. Finally, a local spectral transform is used to measure, directly from the numerical data, the wave-train characteristics of the shock profile.