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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

Brenton LeMesurier 1,

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.
Keywords: discrete NLS, long wave approximation, lattice wave equations, Airy PDE, Binary waveguide arrays, third derivative NLS., optical lattices.
Mathematics Subject Classification: Primary: 34A33; Secondary: 78M34, 65L0.
Citation: Brenton LeMesurier. Continuum approximations for pulses generated by impulsive initial data in binary exciton chain systems. Discrete and 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 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.

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 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.

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