About this Journal Submit a Manuscript Table of Contents
International Journal of Differential Equations
Volume 2012 (2012), Article ID 720687, 17 pages
http://dx.doi.org/10.1155/2012/720687
Research Article

Surfaces of a Constant Negative Curvature

1Mathematics Department, College of Science and Information Technology, Zarqa Private University, Zarqa, Jordan
2Department of Mathematics, Faculty of Science, University of Tabouk, Ministry of Higher Education, P.O. Box 1144, Tabouk, Saudi Arabia

Received 10 March 2012; Revised 30 May 2012; Accepted 13 August 2012

Academic Editor: Wenming Zou

Copyright © 2012 G. M. Gharib. 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, Cambridge University Press, Cambridge, UK, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. S. S. Chern and K. Tenenblat, “Pseudospherical surfaces and evolution equations,” Studies in Applied Mathematics, vol. 74, no. 1, pp. 55–83, 1986. View at Zentralblatt MATH
  3. M. Wadati, H. Sanuki, and K. Konno, “Relationships among inverse method, Bäcklund transformation and an infinite number of conservation laws,” Progress of Theoretical Physics, vol. 53, pp. 419–436, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,” Studies in Applied Mathematics, vol. 53, no. 4, pp. 249–315, 1974. View at Zentralblatt MATH
  5. C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the korteweg-de Vries equation,” Physics Letters, vol. 19, pp. 1095–1097, 1967.
  6. R. Sasaki, “Soliton equations and pseudospherical surfaces,” Nuclear Physics B, vol. 154, no. 2, pp. 343–357, 1979. View at Publisher · View at Google Scholar
  7. A. H. Khater, D. K. Callebaut, and S. M. Sayed, “Conservation laws for some nonlinear evolution equations which describe pseudo-spherical surfaces,” Journal of Geometry and Physics, vol. 51, no. 3, pp. 332–352, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. A. H. Khater, A. A. Abdalla, A. M. Shehatah, D. K. Callebaut, and S. M. Sayed, “Bäcklund transformations and exact solutions for self-dual SU(3) Yang-Mills fields,” Il Nuovo Cimento della Società Italiana di Fisica B, vol. 114, no. 1, pp. 1–10, 1999.
  9. A. H. Khater, D. K. Callebaut, and O. H. El-Kalaawy, “Bäcklund transformations and exact soliton solutions for nonlinear Schrödinger-type equations,” Il Nuovo Cimento della Società Italiana di Fisica B, vol. 113, no. 9, pp. 1121–1136, 1998.
  10. A. H. Khater, D. K. Callebaut, and R. S. Ibrahim, “Bäcklund transformations and Painlevé analysis: exact soliton solutions for the unstable nonlinear Schrödinger equation modeling electron beam plasma,” Physics of Plasmas, vol. 5, no. 2, pp. 395–400, 1998. View at Publisher · View at Google Scholar
  11. A. H. Khater, D. K. Callebaut, and A. R. Seadawy, “Nonlinear dispersive instabilities in Kelvin-Helmholtz magnetohydrodynamic flow,” Physica Scripta, vol. 67, pp. 340–349, 2003.
  12. A. H. Khater, O. H. El-Kalaawy, and D. K. Callebaut, “Bäcklund transformations for Alfvén solitons in a relativistic electron-positron plasma,” Physica Scripta, vol. 58, pp. 545–548, 1998.
  13. K. Chadan and P. C. Sabatier, Inverse Problems in Quantum Scattering Theory, Springer, New York, NY, USA, 1977.
  14. A. H. Khater, W. Malfliet, D. K. Callebaut, and E. S. Kamel, “Travelling wave solutions of some classes of nonlinear evolution equations in (1+1) and (2+1) dimensions,” Journal of Computational and Applied Mathematics, vol. 140, no. 1-2, pp. 469–477, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. K. Konno and M. Wadati, “Simple derivation of Bäcklund transformation from Riccati form of inverse method,” Progress of Theoretical Physics, vol. 53, no. 6, pp. 1652–1656, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. M. Marvan, “Scalar second-order evolution equations possessing an irreducible sl2-valued zero-curvature representation,” Journal of Physics A, vol. 35, no. 44, pp. 9431–9439, 2002. View at Publisher · View at Google Scholar
  17. R. M. Miura, Bäcklund Transformations, the Inverse Scattering Method, Solitons, and Their Applications, vol. 515 of Lecture Notes in Mathematics, Springer, New York, NY, USA, 1976.
  18. E. G. Reyes, “Pseudo-spherical surfaces and integrability of evolution equations,” Journal of Differential Equations, vol. 147, no. 1, pp. 195–230, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. E. G. Reyes, “Conservation laws and Calapso-Guichard deformations of equations describing pseudo-spherical surfaces,” Journal of Mathematical Physics, vol. 41, no. 5, pp. 2968–2989, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. A. V. Shchepetilov, “The geometric sense of the Sasaki connection,” Journal of Physics A, vol. 36, no. 13, pp. 3893–3898, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self–modulation of waves in nonlinear Media,” Soviet Physics, vol. 34, pp. 62–69, 1972.
  22. J. A. Cavalcante and K. Tenenblat, “Conservation laws for nonlinear evolution equations,” Journal of Mathematical Physics, vol. 29, no. 4, pp. 1044–1049, 1988. View at Publisher · View at Google Scholar
  23. R. Beals, M. Rabelo, and K. Tenenblat, “Bäcklund transformations and inverse scattering solutions for some pseudospherical surface equations,” Studies in Applied Mathematics, vol. 81, no. 2, pp. 125–151, 1989. View at Zentralblatt MATH
  24. M. Crampin, “Solitons and SL(2,R),” Physics Letters A, vol. 66, no. 3, pp. 170–172, 1978. View at Publisher · View at Google Scholar
  25. A. C. Scott, F. Y. F. Chu, and D. W. McLaughlin, “The soliton: a new concept in applied science,” Proceedings of the IEEE, vol. 61, pp. 1443–1483, 1973.
  26. E. G. Reyes, “On Geometrically integrable equations and Hierarchies of pseudospherical type,” Contemporary Mathematics, vol. 285, pp. 145–155, 2001.
  27. M. L. Rabelo and K. Tenenblat, “A classification of pseudospherical surface equations of type ut=uxxx+G(u,ux,uxx),” Journal of Mathematical Physics, vol. 33, no. 2, pp. 537–549, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. A. Sakovich and S. Sakovich, “On transformations of the Rabelo equations,” SIGMA. Symmetry, Integrability and Geometry, vol. 3, pp. 1–8, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. M. J. Ablowitz, R. Beals, and K. Tenenblat, “On the solution of the generalized wave and generalized sine-Gordon equations,” Studies in Applied Mathematics, vol. 74, no. 3, pp. 177–203, 1986. View at Zentralblatt MATH
  30. S. M. Sayed, O. O. Elhamahmy, and G. M. Gharib, “Travelling wave solutions for the KdV-Burgers-Kuramoto and nonlinear Schrödinger equations which describe pseudospherical surfaces,” Journal of Applied Mathematics, vol. 2008, Article ID 576783, 10 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. S. M. Sayed, A. M. Elkholy, and G. M. Gharib, “Exact solutions and conservation laws for Ibragimov-Shabat equation which describe pseudo-spherical surface,” Computational & Applied Mathematics, vol. 27, no. 3, pp. 305–318, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. K. Tenenblat, Transformations of Manifolds and Applications to Differential Equations, vol. 93 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Addison Wesley Longman, Harlow, UK, 1998.
  33. M. C. Nucci, “Pseudopotentials, Lax equations and Bäcklund transformations for nonlinear evolution equations,” Journal of Physics A, vol. 21, no. 1, pp. 73–79, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  34. S. M. Sayed and G. M. Gharib, “Canonical reduction of self-dual Yang-Mills equations to Fitzhugh-Nagumo equation and exact solutions,” Chaos, Solitons and Fractals, vol. 39, no. 2, pp. 492–498, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH