Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2017, Article ID 5157123, 6 pages
https://doi.org/10.1155/2017/5157123
Research Article

Bäcklund Transformations between the KdV Equation and a New Nonlinear Evolution Equation

School of Mathematical Sciences, Yangzhou University, Yangzhou, China

Correspondence should be addressed to Xifang Cao; nc.ude.uzy@oacfx

Received 10 January 2017; Accepted 19 February 2017; Published 2 March 2017

Academic Editor: Alessandro Arsie

Copyright © 2017 Xifang Cao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, vol. 149 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, UK, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  2. N. J. Zabusky and M. D. Kruskal, “Interaction of ‘solitons’ in a collisionless plasma and the recurrence of initial states,” Physical Review Letters, vol. 15, no. 6, pp. 240–243, 1965. View at Publisher · View at Google Scholar · View at Scopus
  3. C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for Solving the Korteweg-deVries Equation,” Physical Review Letters, vol. 19, no. 19, pp. 1095–1097, 1967. View at Publisher · View at Google Scholar
  4. V. B. Matveen and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, Germany, 1991.
  5. C. Rogers and W. K. Schief, Bäcklund and Darboux Transformations. Geometry and Modern Applications in Soliton Theory, Cambridge University Press, Cambridge, UK, 2002.
  6. C. Rogers and W. F. Shadwick, Bäcklund Transformations and Their Applications, Academic Press, New York, NY, USA, 1982.
  7. R. M. Miura, “Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation,” Journal of Mathematical Physics, vol. 9, pp. 1202–1204, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. G. L. Lamb Jr., “Bäcklund transformations for certain nonlinear evolution equations,” Journal of Mathematical Physics, vol. 15, pp. 2157–2165, 1974. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. S. I. Svinolupov, V. V. Sokolov, and R. I. Yamilov, “Bäcklund transformations for integrable evolution equations,” Doklady Akademii Nauk SSSR, vol. 271, pp. 802–805, 1983 (Russian). View at Google Scholar
  10. K. Tenenblat, Transformations of manifolds and applications to differential equations, vol. 93 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman, Harlow, UK, 1998. View at MathSciNet
  11. T. Chou, “Bäcklund transformation of nonlinear evolution equations,” Acta Mathematicae Applicatae Sinica, vol. 2, no. 1, pp. 87–94, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  12. H. Wu, “On Bäcklund transformations for nonlinear partial differential equations,” Journal of Mathematical Analysis and Applications, vol. 192, no. 1, pp. 151–179, 1995. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. X. Cao, “Bäcklund transformations with two pseudo-potential,” Applied Mathematics Letters, vol. 61, pp. 13–19, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. H. D. Wahlquist and F. B. Estabrook, “Bäcklund transformation for solutions of the Korteweg-de Vries equation,” Physical Review Letters, vol. 31, pp. 1386–1390, 1973. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. H. D. Wahlquist and F. B. Estabrook, “Prolongation structures of nonlinear evolution equations,” Journal of Mathematical Physics, vol. 16, pp. 1–7, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. X. Cao, C. Xu, and H. Wu, “On Miura transformations among nonlinear partial differential equations,” Journal of Mathematical Physics, vol. 47, no. 8, Article ID 083515, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus