August  2016, 21(6): 1813-1837. doi: 10.3934/dcdsb.2016024

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

1. 

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

Received  April 2015 Revised  February 2016 Published  June 2016

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.
Citation: Brenton LeMesurier. Continuum approximations for pulses generated by impulsive initial data in binary exciton chain systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1813-1837. doi: 10.3934/dcdsb.2016024
References:
[1]

A. B. Aceves, A. Auditore, M. Conforti and C. De Angelis, Discrete localized modes in binary waveguide arrays,, in Nonlinear Photonics (NLP), (2013), 38.  doi: 10.1109/NLP.2013.6646384.  Google Scholar

[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.  doi: 10.1016/j.optcom.2013.01.068.  Google Scholar

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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).  doi: 10.1103/PhysRevB.61.1129.  Google Scholar

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M. Conforti, C. De Angelis and T. R. Akylas, Energy localization and transport in binary waveguide arrays,, Physical Review A, 83 (2011).  doi: 10.1103/PhysRevA.83.043822.  Google Scholar

[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).  doi: 10.1103/PhysRevA.85.063836.  Google Scholar

[6]

M. Creutz and A. Gocksch, Higher order hybrid Monte-Carlo algorithms,, Phys. Rev. Lett., 63 (1989), 9.  doi: 10.1103/PhysRevLett.63.9.  Google Scholar

[7]

A. S. Davydov, Theory of Molecular Excitations,, Plenum press, (1971).   Google Scholar

[8]

A. S. Davydov, Solitons in molecular systems,, Physica Scripta, 20 (1979), 387.  doi: 10.1088/0031-8949/20/3-4/013.  Google Scholar

[9]

A. S. Davydov, Solitons in Molecular Systems,, Kluwer Academic Publishers, (1991).  doi: 10.1007/978-94-011-3340-1.  Google Scholar

[10]

J. Eilbeck, P. Lomdahl and A. Scott, The discrete self-trapping equation,, Physica D: Nonlinear Phenomena, 16 (1985), 318.  doi: 10.1016/0167-2789(85)90012-0.  Google Scholar

[11]

E. Forest, Canonical integrators as tracking codes,, AIP Conference Proceedings, 184 (1989), 1106.   Google Scholar

[12]

I. L. Garanovich, A. A. Sukhorukov and Y. S. Kivshar, Surface multi-gap vector solitons,, Optics Express, 14 (2006), 4780.  doi: 10.1364/OE.14.004780.  Google Scholar

[13]

O. Gonzales, Time integration and discrete Hamiltonian systems,, Journal of Nonlinear Science, 6 (1996), 449.  doi: 10.1007/BF02440162.  Google Scholar

[14]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations,, 2nd edition, (2006).   Google Scholar

[15]

T. Holstein, Studies of polaron motion: Part I. the molecular-crystal model,, Ann. Phys., 8 (1959), 325.   Google Scholar

[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.  doi: 10.1016/j.matcom.2010.11.017.  Google Scholar

[17]

B. LeMesurier, Energetic pulses in exciton-phonon molecular chains, and conservative numerical methods for quasi-linear Hamiltonian systems,, Physical Review E, 88 (2013).   Google Scholar

[18]

R. I. McLachlin, On the numerical integration of ordinary differential equations by symmetric composition methods,, SIAM Journal of Scientific Computation, 16 (1995), 151.  doi: 10.1137/0916010.  Google Scholar

[19]

D. Pelinovsky and V. Rothos, Bifurcations of travelling wave solutions in the discrete NLS equation,, Physica D, 202 (2005), 16.  doi: 10.1016/j.physd.2005.01.016.  Google Scholar

[20]

A. C. Scott, The vibrational structure of Davydov solitons,, Physica Scripta, 25 (1982), 651.  doi: 10.1088/0031-8949/25/5/015.  Google Scholar

[21]

A. C. Scott, Davydov's soliton,, Physics Reports, 217 (1992), 1.   Google Scholar

[22]

A. A. Sukhorukov and Y. S. Kivshar, Discrete gap solitons in modulated waveguide arrays,, Opt. Lett., 27 (2002), 2112.  doi: 10.1364/NLGW.2002.NLTuA2.  Google Scholar

[23]

M. Suzuki, Fractal decomposition of exponential operators with applications to many-body theories and Monte-Carlo simulations,, Phys. Lett. A, 146 (1990), 319.  doi: 10.1016/0375-9601(90)90962-N.  Google Scholar

[24]

H. Yoshida, Construction of higher order symplectic integrators,, Phys. Lett. A, 150 (1990), 262.  doi: 10.1016/0375-9601(90)90092-3.  Google Scholar

show all references

References:
[1]

A. B. Aceves, A. Auditore, M. Conforti and C. De Angelis, Discrete localized modes in binary waveguide arrays,, in Nonlinear Photonics (NLP), (2013), 38.  doi: 10.1109/NLP.2013.6646384.  Google Scholar

[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.  doi: 10.1016/j.optcom.2013.01.068.  Google Scholar

[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).  doi: 10.1103/PhysRevB.61.1129.  Google Scholar

[4]

M. Conforti, C. De Angelis and T. R. Akylas, Energy localization and transport in binary waveguide arrays,, Physical Review A, 83 (2011).  doi: 10.1103/PhysRevA.83.043822.  Google Scholar

[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).  doi: 10.1103/PhysRevA.85.063836.  Google Scholar

[6]

M. Creutz and A. Gocksch, Higher order hybrid Monte-Carlo algorithms,, Phys. Rev. Lett., 63 (1989), 9.  doi: 10.1103/PhysRevLett.63.9.  Google Scholar

[7]

A. S. Davydov, Theory of Molecular Excitations,, Plenum press, (1971).   Google Scholar

[8]

A. S. Davydov, Solitons in molecular systems,, Physica Scripta, 20 (1979), 387.  doi: 10.1088/0031-8949/20/3-4/013.  Google Scholar

[9]

A. S. Davydov, Solitons in Molecular Systems,, Kluwer Academic Publishers, (1991).  doi: 10.1007/978-94-011-3340-1.  Google Scholar

[10]

J. Eilbeck, P. Lomdahl and A. Scott, The discrete self-trapping equation,, Physica D: Nonlinear Phenomena, 16 (1985), 318.  doi: 10.1016/0167-2789(85)90012-0.  Google Scholar

[11]

E. Forest, Canonical integrators as tracking codes,, AIP Conference Proceedings, 184 (1989), 1106.   Google Scholar

[12]

I. L. Garanovich, A. A. Sukhorukov and Y. S. Kivshar, Surface multi-gap vector solitons,, Optics Express, 14 (2006), 4780.  doi: 10.1364/OE.14.004780.  Google Scholar

[13]

O. Gonzales, Time integration and discrete Hamiltonian systems,, Journal of Nonlinear Science, 6 (1996), 449.  doi: 10.1007/BF02440162.  Google Scholar

[14]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations,, 2nd edition, (2006).   Google Scholar

[15]

T. Holstein, Studies of polaron motion: Part I. the molecular-crystal model,, Ann. Phys., 8 (1959), 325.   Google Scholar

[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.  doi: 10.1016/j.matcom.2010.11.017.  Google Scholar

[17]

B. LeMesurier, Energetic pulses in exciton-phonon molecular chains, and conservative numerical methods for quasi-linear Hamiltonian systems,, Physical Review E, 88 (2013).   Google Scholar

[18]

R. I. McLachlin, On the numerical integration of ordinary differential equations by symmetric composition methods,, SIAM Journal of Scientific Computation, 16 (1995), 151.  doi: 10.1137/0916010.  Google Scholar

[19]

D. Pelinovsky and V. Rothos, Bifurcations of travelling wave solutions in the discrete NLS equation,, Physica D, 202 (2005), 16.  doi: 10.1016/j.physd.2005.01.016.  Google Scholar

[20]

A. C. Scott, The vibrational structure of Davydov solitons,, Physica Scripta, 25 (1982), 651.  doi: 10.1088/0031-8949/25/5/015.  Google Scholar

[21]

A. C. Scott, Davydov's soliton,, Physics Reports, 217 (1992), 1.   Google Scholar

[22]

A. A. Sukhorukov and Y. S. Kivshar, Discrete gap solitons in modulated waveguide arrays,, Opt. Lett., 27 (2002), 2112.  doi: 10.1364/NLGW.2002.NLTuA2.  Google Scholar

[23]

M. Suzuki, Fractal decomposition of exponential operators with applications to many-body theories and Monte-Carlo simulations,, Phys. Lett. A, 146 (1990), 319.  doi: 10.1016/0375-9601(90)90962-N.  Google Scholar

[24]

H. Yoshida, Construction of higher order symplectic integrators,, Phys. Lett. A, 150 (1990), 262.  doi: 10.1016/0375-9601(90)90092-3.  Google Scholar

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