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The phenomenon of coherent energetic pulse propagation in exciton-phononmolecular chains such as $\alpha$-helix protein is studied using an ODE systemmodel of Davydov-Scott type, both with numerical studies using a newunconditionally stable fourth order accurate energy-momentum conserving timediscretization, and with analytical explanation of the main numericalobservations.  Physically natural impulsive initial data associated with the energy releasedby ATP hydrolysis are used, and the best current estimates of physicalparameter values. In contrast to previous studies based on a proposed long waveapproximation by the nonlinear Schr\"odinger (NLS) equation and focusing oninitial data resembling the soliton solutions of that equation, the resultshere instead lead to approximation by the third derivative nonlinearSchr\"odinger equation, giving a far better fit to observed behavior. A goodpart of the behavior is indeed explained well by the linear part of thatequation, the Airy PDE, while other significant features do not fit any PDEapproximation, but are instead explained well by a linearized analysis of theODE system.  A convenient method is described for construction the highly stable, accurateconservative time discretizations used, with proof of its desirable propertiesfor a large class of Hamiltonian systems, including a variety of molecularmodels.

Energetic Pulses in Exciton-Phonon Molecular Chains, and Conservative Numerical Methods for Quasi-linear Hamiltonian Systems