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Formation, control, and dynamics of N localized structures in the Peyrard-Bishop model

Elías Zamora-Sillero1,*, A. V. Shapovalov2,3,†, and Francisco J. Esteban4,‡

  • 1Departamento de Fisica Aplicada I, E.U.P. Universidad de Sevilla Virgen de Africa 7, 41011 Sevilla, Spain
  • 2Laboratory of Mathematical Physics, Tomsk Polytechnical University, 30th Lenin Avenue, 634050 Tomsk, Russia
  • 3Department of Theoretical Physics, Tomsk State University, 36th Lenin Avenue, 634050 Tomsk, Russia
  • 4Cell Biology Section, Department of Experimental Biology, University of Jaen, Jaen, Spain

  • *ezamora80@us.es
  • †shpv@phys.tsu.ru
  • ‡festeban@ujaen.es
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Phys. Rev. E 76, 066603 – Published 6 December, 2007

DOI: https://doi.org/10.1103/PhysRevE.76.066603

Abstract

We explore in detail the creation of stable localized structures in the form of localized energy distributions that arise from general initial conditions in the Peyrard-Bishop (PB) model. By means of a method based on the inverse scattering transform we study the solutions of PB model equations obtained in the form of planar waves whose amplitudes are described by the nonlinear Schrödinger equation (NLS). For localized initial conditions different from the pure N-soliton shape, we have obtained analytical results that predict and control the number, amplitude, and velocity of the NLS solitary waves. To verify the validity of these results we have carried out numerical simulations of the PB model with the use of realistic values of parameters and the initial conditions in the form of planar waves whose modulated amplitudes are given by the examples studied in the NLS. In the simulations we have found that N localized structures arise in agreement with the prediction of the analytical results obtained in the NLS.

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  57. For a general initial condition F1(y,0), a soliton solution is possible if at least one point in the discrete spectrum exists among all the eigenvalue problem solutions (18). It can be shown [35] that it implies the neccesary condition S>ln(2+3)≃1.32. We can see in our examples that the threshold value S0 is greater than 1.32 and depends on the concrete initial condition.

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Phys. Rev. E 76, 066603– Published 6 December, 2007
  • Received 2 June 2007

© 2007 American Physical Society

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