Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 020, 9 pages      arXiv:1710.06091      https://doi.org/10.3842/SIGMA.2018.020
Contribution to the Special Issue on Symmetries and Integrability of Difference Equations

Special Solutions of Bi-Riccati Delay-Differential Equations

Bjorn K. Berntson
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK

Received May 15, 2017, in final form March 02, 2018; Published online March 09, 2018

Abstract
Delay-differential equations are functional differential equations that involve shifts and derivatives with respect to a single independent variable. Some integrability candidates in this class have been identified by various means. For three of these equations we consider their elliptic and soliton-type solutions. Using Hirota's bilinear method, we find that two of our equations possess three-soliton-type solutions.

Key words: delay-differential equations; elliptic solutions; solitons.

pdf (280 kb)   tex (13 kb)

References

  1. Ablowitz M.J., Satsuma J., Solitons and rational solutions of nonlinear evolution equations, J. Math. Phys. 19 (1978), 2180-2186.
  2. Babalic C.N., Carstea A.S., On some new forms of lattice integrable equations, Cent. Eur. J. Phys. 12 (2014), 341-347, arXiv:1508.04998.
  3. Berntson B.K., Integrable delay-differential equations, Ph.D. Thesis, University College London, 2017.
  4. Berntson B.K., Halburd R., Integrable delay-differential equations, in preparation.
  5. Brezhnev Y., Leble S., On integration of the closed KdV dressing chain, math-ph/0502052.
  6. Carstea A.S., Bilinear approach to delay-Painlevé equations, J. Phys. A: Math. Theor. 44 (2011), 105202, 7 pages.
  7. Dai C.Q., Cen X., Wu S.S., Exact travelling wave solutions of the discrete sine-Gordon equation obtained via the exp-function method, Nonlinear Anal. 70 (2009), 58-63.
  8. Grammaticos B., Ramani A., Moreira I.C., Delay-differential equations and the Painlevé transcendents, Phys. A 196 (1993), 574-590.
  9. Hietarinta J., Introduction to the Hirota bilinear method, in Integrability of Nonlinear Systems (Pondicherry, 1996), Lecture Notes in Phys., Vol. 495, Springer, Berlin, 1997, 95-103, solv-int/9708006.
  10. Hille E., Analytic function theory, Vol. I, AMS Chelsea Publishing Series, Vol. 269, Amer. Math. Soc., Providence, RI, 2012.
  11. Joshi N., Direct `delay' reductions of the Toda equation, J. Phys. A: Math. Theor. 42 (2009), 022001, 8 pages, arXiv:0810.5581.
  12. Quispel G.R.W., Capel H.W., Sahadevan R., Continuous symmetries of differential-difference equations: the Kac-van Moerbeke equation and Painlevé reduction, Phys. Lett. A 170 (1992), 379-383.
  13. Quispel G.R.W., Roberts J.A.G., Thompson C.J., Integrable mappings and soliton equations, Phys. Lett. A 126 (1988), 419-421.
  14. Ramani A., Grammaticos B., Tamizhmani K.M., Painlevé analysis and singularity confinement: the ultimate conjecture, J. Phys. A: Math. Gen. 26 (1993), L53-L58.
  15. vom Hofe R., Grundvorstellungen mathematischer Inhalte, Texte zur Didaktik der Mathematik, Spektrum Akademischer Verlag GmbH, Heidelberg, 1995.

Previous article  Next article   Contents of Volume 14 (2018)