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

TL;DR

This paper investigates energetic pulse propagation in exciton-phonon chains using a Davydov-Scott model, introducing a new stable numerical method and revealing that the third derivative nonlinear Schrödinger equation better describes observed behaviors than the classical NLS approximation.

Contribution

It presents a novel stable, energy-momentum conserving numerical scheme for Hamiltonian systems and demonstrates that the third derivative NLS equation more accurately models pulse dynamics in molecular chains.

Findings

Pulse behavior aligns with the third derivative NLS equation.

Linear analysis of the ODE system explains significant features.

A new stable numerical discretization method is developed and validated.

Abstract

The phenomenon of coherent energetic pulse propagation in exciton-phonon molecular chains such as $\alpha$-helix protein is studied using an ODE system model of Davydov-Scott type, both with numerical studies using a new unconditionally stable fourth order accurate energy-momentum conserving time discretization, and with analytical explanation of the main numerical observations. Physically natural impulsive initial data associated with the energy released by ATP hydrolysis are used, and the best current estimates of physical parameter values. In contrast to previous studies based on a proposed long wave approximation by the nonlinear Schr\"odinger (NLS) equation and focusing on initial data resembling the soliton solutions of that equation, the results here instead lead to approximation by the third derivative nonlinear Schr\"odinger equation, giving a far better fit to observed behavior. A good part of the behavior is indeed explained well by the linear part of that equation, the Airy PDE, while other significant features do not fit any PDE approximation, but are instead explained well by a linearized analysis of the ODE system. A convenient method is described for construction the highly stable, accurate conservative time discretizations used, with proof of its desirable properties for a large class of Hamiltonian systems, including a variety of molecular models.