Numerical studies of solitons on lattices

  • J. C. Eilbeck
Part III: Lattice Excitations and Localised Modes
Part of the Lecture Notes in Physics book series (LNP, volume 393)


We use path-following methods and spectral collocation methods to study families of solitary wave solutions of lattice equations. These techniques are applied to a number of 1-D and 2-D lattices, including an electrical lattice introduced by Remoissenet and co-workers, and a 2-D lattice suggested by Zakharov, which in a particular continuum limit reduces to the Kadomtsev-Petviashvili equation.


Solitary Wave Continuum Limit Lattice Equation Solitary Wave Solution Toda Lattice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. [1]
    M Toda. Theory of Nonlinear Lattices. Springer, Berlin, 1981.Google Scholar
  2. [2]
    M J Ablowitz and J F Ladik. A nonlinear difference scheme and inverse scattering. Stud. Appl. Math., 55:213–229, 1976.Google Scholar
  3. [3]
    J C Eilbeck and R Flesch. Calculation of families of solitary waves on discrete lattices. Phys. Lett. A, 149:200–202, 1990.Google Scholar
  4. [4]
    C Canuto, M Y Hussaini, A Quarteroni, and T A Zang. Spectral Methods in Fluid Mechanics. Springer-Verlag, Berlin, 1988.Google Scholar
  5. [5]
    R Seydel. From Equilibrium to Chaos — Practical Bifurcation and Stability Analysis. Elsevier, London, 1988.Google Scholar
  6. [6]
    D Hochstrasser, F G Mertens, and H Büttner. An iterative method for the calculation of narrow solitary excitations on atomic chains. Physica, D35:259–266, 1989.Google Scholar
  7. [7]
    D B Duncan, C H Walshaw, and J A D Wattis. A symplectic solver for lattice equations. (these proceedings), 1991.Google Scholar
  8. [8]
    M Remoissenet and B Michaux. Electrical transmission lines and soliton propagation in physical systems. In G. Maugin, editor, Continuum Models and Discrete Systems, Proceedings of the CMDS6 Conference, Dijon 1989, London, 1990. Longman.Google Scholar
  9. [9]
    P L Christiansen and A C Scott. Davydov's Soliton Revisited. Plenum, New York, 1990.Google Scholar
  10. [10]
    J C Eilbeck, P S Lomdahl, and A C Scott. The discrete self-trapping equation. Physica D: Nonlinear Phenomena, 16:318–338, 1985.Google Scholar
  11. [11]
    H Feddersen. Solitary wave solutions to the discrete nonlinear Schrödinger equation. (these proceedings), 1991.Google Scholar
  12. [12]
    O A Druzhinin and L A Ostrovskii. Solitons in discrete lattices. Preprint, to be published in Phys. Lett. A, 1991.Google Scholar
  13. [13]
    D B Duncan, J C Eilbeck, C H Walshaw, and V E Zakharov. Solitary waves on a strongly anisotropic KP lattice. Submitted to Phys. Lett. A., 1991.Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • J. C. Eilbeck
    • 1
  1. 1.Department of MathematicsHeriot-Watt UniversityEdinburghUK

Personalised recommendations