Continuum approximations for pulses generated by impulsive initial data in binary exciton chain systems

Brenton  LeMesurier

doi:10.3934/dcdsb.2016024

Article Contents

2016, Volume 21, Issue 6: 1813-1837. Doi: 10.3934/dcdsb.2016024

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Continuum approximations for pulses generated by impulsive initial data in binary exciton chain systems

Brenton LeMesurier^1,

1.
Department of Mathematics, College of Charleston, 66 George St. Charleston, SC 29424

Received: March 31, 2015

Revised: January 31, 2016

Published: July 2016

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Abstract

Systems of ODE's with forms related to the discrete nonlinear Schrödinger equation arise in a variety of semi-classical molecular models, such as models of polymers with coupling between nearby excitable states, and reductions of quantum many-body systems. Similar systems also arise as models of arrays of nonlinear optical wave-guides, which can involve new features such as binary alternation of coefficients: so-called Binary Exciton Chain Systems.
Solutions of such systems are often seen to develop regions of slow variation even when the initial data are impulsive: in particular the emergence of a slowly varying leading pulse that propagates in an approximately traveling wave form, and in stationary oscillations near the endpoints. This has motivated the search for long-wave approximations by PDEs.
In this article it is observed that the patterns of slow variation are substantially different from those assumed in some previously-considered long-wave approximations, and several new PDE approximations are presented: third order systems describing leading pulses of approximately traveling wave form (related to the Airy PDE in the linearized case), and a quite different system describing stationary oscillations near an endpoint with zero boundary conditions. Numerical solutions of these PDE models and some linear analysis confirm that they provide a good agreement with the long-wave phenomena observed in the ODE systems.

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Mathematics Subject Classification: Primary: 34A33; Secondary: 78M34, 65L04.

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