Article Contents
Article Contents

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

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

 Citation:

•  [1] A. B. Aceves, A. Auditore, M. Conforti and C. De Angelis, Discrete localized modes in binary waveguide arrays, in Nonlinear Photonics (NLP), 2013 IEEE International Workshop, 2013, 38-42.doi: 10.1109/NLP.2013.6646384. [2] A. Auditore, M. Conforti, C. De Angelis and A. B. Aceves, Dark-antidark solitons in waveguide arrays with alternating positive-negative couplings, Optics Communications, 297 (2013), 125-128, URL http://www.sciencedirect.com/science/article/pii/S0030401813001545.doi: 10.1016/j.optcom.2013.01.068. [3] L. Brizhik, A. Eremko, L. Cruzeiro-Hansson and Y. Olkhovska, Soliton dynamics and Peierls-Nabarro barrier in a discrete molecular chain, Physical Review B, 61 (2000), p1129.doi: 10.1103/PhysRevB.61.1129. [4] M. Conforti, C. De Angelis and T. R. Akylas, Energy localization and transport in binary waveguide arrays, Physical Review A, 83 (2011), 043822.doi: 10.1103/PhysRevA.83.043822. [5] M. Conforti, C. De Angelis, T. R. Akylas and A. B. Aceves, Modulational stability and gap solitons of gapless systems: Continuous versus discrete, Physical Review A, 85 (2012), 063836.doi: 10.1103/PhysRevA.85.063836. [6] M. Creutz and A. Gocksch, Higher order hybrid Monte-Carlo algorithms, Phys. Rev. Lett., 63 (1989), 9-12.doi: 10.1103/PhysRevLett.63.9. [7] A. S. Davydov, Theory of Molecular Excitations, Plenum press, New York, 1971. [8] A. S. Davydov, Solitons in molecular systems, Physica Scripta, 20 (1979), 387-394.doi: 10.1088/0031-8949/20/3-4/013. [9] A. S. Davydov, Solitons in Molecular Systems, Kluwer Academic Publishers, Dordrecht, 1991.doi: 10.1007/978-94-011-3340-1. [10] J. Eilbeck, P. Lomdahl and A. Scott, The discrete self-trapping equation, Physica D: Nonlinear Phenomena, 16 (1985), 318-338, URL http://www.sciencedirect.com/science/article/pii/0167278985900120.doi: 10.1016/0167-2789(85)90012-0. [11] E. Forest, Canonical integrators as tracking codes, AIP Conference Proceedings, 184 (1989), 1106-1136. [12] I. L. Garanovich, A. A. Sukhorukov and Y. S. Kivshar, Surface multi-gap vector solitons, Optics Express, 14 (2006), 4780-4785.doi: 10.1364/OE.14.004780. [13] O. Gonzales, Time integration and discrete Hamiltonian systems, Journal of Nonlinear Science, 6 (1996), 449-467.doi: 10.1007/BF02440162. [14] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations, 2nd edition, Springer, 2006. [15] T. Holstein, Studies of polaron motion: Part I. the molecular-crystal model, Ann. Phys., 8 (1959), 325-389. [16] B. LeMesurier, Studying Davydov's ODE model of wave motion in alpha-helix protein using exactly energy-momentum conserving discretizations for Hamiltonian systems, Mathematics and Computers in Simulation, 82 (2012), 1239-1248, Published online 30 December 2010.doi: 10.1016/j.matcom.2010.11.017. [17] B. LeMesurier, Energetic pulses in exciton-phonon molecular chains, and conservative numerical methods for quasi-linear Hamiltonian systems, Physical Review E, 88 (2013), 032707, URL http://arxiv.org/abs/1301.2996. [18] R. I. McLachlin, On the numerical integration of ordinary differential equations by symmetric composition methods, SIAM Journal of Scientific Computation, 16 (1995), 151-168.doi: 10.1137/0916010. [19] D. Pelinovsky and V. Rothos, Bifurcations of travelling wave solutions in the discrete NLS equation, Physica D, 202 (2005), 16-36.doi: 10.1016/j.physd.2005.01.016. [20] A. C. Scott, The vibrational structure of Davydov solitons, Physica Scripta, 25 (1982), 651-658.doi: 10.1088/0031-8949/25/5/015. [21] A. C. Scott, Davydov's soliton, Physics Reports, 217 (1992), 1-67. [22] A. A. Sukhorukov and Y. S. Kivshar, Discrete gap solitons in modulated waveguide arrays, Opt. Lett., 27 (2002), 2112-2114.doi: 10.1364/NLGW.2002.NLTuA2. [23] M. Suzuki, Fractal decomposition of exponential operators with applications to many-body theories and Monte-Carlo simulations, Phys. Lett. A, 146 (1990), 319-323.doi: 10.1016/0375-9601(90)90962-N. [24] H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), 262-268.doi: 10.1016/0375-9601(90)90092-3.