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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Bäcklund Transformation and Quasi-Integrable Deformation of Mixed Fermi-Pasta-Ulam and Frenkel-Kontorova Models

Discontinuity, Nonlinearity, and Complexity 7(1) (2018) 31--41 | DOI:10.5890/DNC.2018.03.003

Kumar Abhinav$^{1}$, A Ghose Choudhury$^{2}$, Partha Guha$^{1}$

$^{1}$ SN Bose National Centre for Basic Sciences JD Block, Sector III, Salt Lake, Kolkata 700106, India

$^{2}$ Department of Physics, Surendranath College, 24/2 Mahatma Gandhi Road, Calcutta 700009, India

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In this paper we study a non-linear partial differential equation (PDE), proposed by Kudryashov [arXiv:1611.06813v1[nlin.SI]], using continuum limit approximation of mixed Fermi-Pasta-Ulam and Frenkel-Kontorova Models. This generalized semi-discrete equation can be considered as a model for the description of non-linear dislocation waves in crystal lattice and the corresponding continuous system can be called mixed generalized potential KdV and sine-Gordon equation. We obtain the Bäcklund transformation of this equation in Riccati form in inverse method. We further study the quasi-integrable deformation of this model.


The authors are grateful to Professors Luiz. A. Ferreira, Wojtek J. Zakrzewski and Betti Hartmann for their encouragement, various useful discussions and critical reading of the draft. This paper is dedicated to the memory of our friend Anjan Kundu, his death cut short a productive career.


  1. [1]  Kudryashov, N.A., Integrable model of nonlinear dislocations, arXiv:1611.06813v1[nlin.SI].
  2. [2]  Kontorova, T.A. and Frenkel, Y.I. (1938), On theory of plastic deformation, JETP, 8, 89, 1340, 1349 (in Russian).
  3. [3]  Braun, O.M. and Kivshar, Y.S. (1998), Nonlinear dynamics of the Frenkel- Kontorova model, Physics Reports, 306, 1-108.
  4. [4]  Fermi, E., Pasta, J.R., and Ulam, S. (Report LA-1940, 1955), Studies of nonlinear problems, Los Alamos: Los Alamos Scientific Laboratory.
  5. [5]  Porter, M.A., Zabusky, N.J., Hu, B., and Campbell, D.K. (2009), Fermi, Pasta, Ulam and the Birth of Experimental Mathematics, American Scientist, 97(3), 214-221. doi:10.1511/2009.78.214.
  6. [6]  Zabusky, N.J. and Kruskal, M.D. (1965), Interactions of solitons in a collisionless plasma and the recurrence of initial states, Physical Review Letters, 15(6), 240-243.
  7. [7]  Faddeev, L.D. and Takhtajan, L.A. (1987), Hamiltonian Methods in the Theory of Solitons. Berlin: Springer-Verlag.
  8. [8]  Das, A. (1989), Integrable models, World Scientific, Singapore.
  9. [9]  Konno, K. and Wadati, M., Simple derivation of Bäcklund transformation from Riccati form of inverse method, Progress of Theoretical Physics, 53(6), 1652-1656.
  10. [10]  Wadati, M., Sanuki, H., and Konno, K., Relationships among inverse method, B¨acklund transformation and an infinite number of conservation laws, Progress of Theoretical Physics, 53(2), 419-436.
  11. [11]  Ferreira, S., Girardello, L., and Sciuto, S. (1978), An infinite set of conservation laws of the supersymmetric sinegordon theory, Phys. Lett. B, 76 303.
  12. [12]  Ferreira, L.A. and Zakrzewski,W.J. (2011), The concept of quasi-integrability: a concrete example, JHEP, 05(130).
  13. [13]  Ferreira, L.A., Luchini, G., and Zakrzewski, W.J. (2013), The concept of Quasi-integrability, Nonlinear and Modern Mathematical Physics AIP Conf. Proc., 1562(43).
  14. [14]  Abhinav, K. and Guha, P. Quasi-Integrability of The KdV System, arXiv:1612.07499[math-ph].
  15. [15]  Abhinav, K. and Guha, P. (2016), Quasi-Integrability in Supersymmetric Sine-Gordon Models, EPL 116 10004.