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Abstract and Applied Analysis
Volume 2014, Article ID 424059, 11 pages
http://dx.doi.org/10.1155/2014/424059
Research Article

Bäcklund Transformation and Quasi-Periodic Solutions for a Variable-Coefficient Integrable Equation

College of Science, China University of Mining and Technology, Xuzhou 221116, China

Received 12 March 2014; Accepted 27 April 2014; Published 18 May 2014

Academic Editor: Tiecheng Xia

Copyright © 2014 Wenjuan Rui and Yufeng Zhang. 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, Cambridge University Press, New York, NY, USA, 1991. View at MathSciNet
  2. V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, Germany, 1991. View at MathSciNet
  3. C. H. Gu, H. S. Hu, and Z. X. Zhou, Darboux Transformation in Solitons Theory and Geometry Applications, Shanghai Science and Technology Press, Shanghai, China, 1999.
  4. T. C. Xia, X. H. Chen, and D. Y. Chen, “Darboux transformation and soliton-like solutions of nonlinear Schrödinger equations,” Chaos, Solitons & Fractals, vol. 26, no. 3, pp. 889–896, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. M. R. Miura, Bäcklund Transformation, Springer, Berlin, Germany, 1978.
  6. Y. Chen, Z. Y. Yan, and H. Q. Zhang, “Exact solutions for a family of variable-coefficient “reaction-duffing” equations via the Bäcklund transformation,” Theoretical and Mathematical Physics, vol. 132, no. 1, pp. 970–975, 2002. View at Publisher · View at Google Scholar · View at Scopus
  7. R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, New York, NY, USA, 2004. View at MathSciNet
  8. A.-M. Wazwaz, “The Hirota's direct method for multiple-soliton solutions for three model equations of shallow water waves,” Applied Mathematics and Computation, vol. 201, no. 1-2, pp. 489–503, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. E. D. Belokolos, A. I. Bobenko, V. Z. Enolskij, A. R. Its, and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer, Berlin, Germany, 1994.
  10. F. Gesztesy and H. Holden, Soliton Equations and Their Algebro-Geometric Solutions—I: (1 + 1)-Dimensional Continuous Models, Cambridge University Press, New York, NY, USA, 2003. View at MathSciNet
  11. F. Gesztesy, H. Holden, J. Michor, and G. Teschl, Soliton Equations and Their Algebro-Geometric Solutions—II: (1 + 1)-Dimensional Discrete Models, Cambridge University Press, Cambridge, New York, NY, USA, 2008. View at MathSciNet
  12. C. Gilson, F. Lambert, J. Nimmo, and R. Willox, “On the combinatorics of the Hirota D-operators,” Proceedings of the Royal Society A: Mathematical, Physical & Engineering Sciences, vol. 452, no. 1945, pp. 223–234, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  13. F. Lambert, I. Loris, and J. Springael, “Classical Darboux transformations and the KP hierarchy,” Inverse Problems, vol. 17, no. 4, pp. 1067–1074, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. F. Lambert and J. Springael, “Soliton equations and simple combinatorics,” Acta Applicandae Mathematicae, vol. 102, no. 2-3, pp. 147–178, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. E. G. Fan, “The integrability of nonisospectral and variable-coefficient KdV equation with binary Bell polynomials,” Physics Letters A, vol. 375, no. 3, pp. 493–497, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Y. C. Hon and E. G. Fan, “Binary Bell polynomial approach to the non-isospectral and variable-coefficient KP equations,” IMA Journal of Applied Mathematics, vol. 77, no. 2, pp. 236–251, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. E. G. Fan and K. W. Chow, “Darboux covariant Lax pairs and infinite conservation laws of the (2+1)-dimensional breaking soliton equation,” Journal of Mathematical Physics, vol. 52, no. 2, Article ID 023504, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  18. E. G. Fan and Y. C. Hon, “Super extension of Bell polynomials with applications to supersymmetric equations,” Journal of Mathematical Physics, vol. 53, no. 1, Article ID 013503, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. E. G. Fan, “New bilinear Bäcklund transformation and Lax pair for the supersymmetric two-Boson equation,” Studies in Applied Mathematics, vol. 127, no. 3, pp. 284–301, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. W. X. Ma, “Bilinear equations, Bell polynomials and linear superposition principle,” Journal of Physics: Conference Series, vol. 411, no. 1, Article ID 012021, 2013. View at Publisher · View at Google Scholar
  21. A. Nakamura, “A direct method of calculating periodic wave solutions to nonlinear evolution equations—I. Exact two-periodic wave solution,” Journal of the Physical Society of Japan, vol. 47, no. 5, pp. 1701–1705, 1979. View at Publisher · View at Google Scholar · View at MathSciNet
  22. A. Nakamura, “A direct method of calculating periodic wave solutions to nonlinear evolution equations—II. Exact one- and two-periodic wave solution of the coupled bilinear equations,” Journal of the Physical Society of Japan, vol. 48, no. 4, pp. 1365–1370, 1980. View at Publisher · View at Google Scholar · View at MathSciNet
  23. Y. C. Hon, E. G. Fan, and Z. Qin, “A kind of explicit quasi-periodic solution and its limit for the TODA lattice equation,” Modern Physics Letters B, vol. 22, no. 8, pp. 547–553, 2008. View at Publisher · View at Google Scholar · View at Scopus
  24. E. G. Fan and Y. C. Hon, “Quasi-periodic waves and asymptotic behavior for Bogoyavlenskiis breaking soliton equation in (2+1) dimensions,” Physical Review E, vol. 78, no. 3, Article ID 036607, 13 pages, 2008. View at Publisher · View at Google Scholar
  25. E. G. Fan, “Quasi-periodic waves and an asymptotic property for the asymmetrical Nizhnik-Novikov-Veselov equation,” Journal of Physics A : Mathematical and Theoretical, vol. 42, no. 9, Article ID 095206, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. W.-X. Ma, R. G. Zhou, and L. Gao, “Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in (2+1) dimensions,” Modern Physics Letters A, vol. 24, no. 21, pp. 1677–1688, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. Y. Zhang, Z. L. Cheng, and X. H. Hong, “Riemann theta function periodic wave solutions for the variable-coefficient mKdV equation,” Chinese Physics B, vol. 21, no. 12, Article ID 120203, 2012. View at Publisher · View at Google Scholar
  28. M. Zamir, The Physics of Pulsatile Flow, Springer, New York, NY, USA, 2000.
  29. F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Reviews of Modern Physics, vol. 71, no. 3, pp. 463–512, 1999. View at Publisher · View at Google Scholar
  30. B. Tian, G. M. Wei, C. Y. Zhang, W. R. Shan, and Y. T. Gao, “Transformations for a generalized variable-coefficient Korteweg-de Vries model from blood vessels, Bose-Einstein condensates, rods and positons with symbolic computation,” Physics Letters A, vol. 356, no. 1, pp. 8–16, 2006. View at Publisher · View at Google Scholar
  31. V. N. Serkin and A. Hasegawa, “Novel soliton solutions of the nonlinear Schrödinger equation model,” Physical Review Letters, vol. 85, no. 21, pp. 4502–4505, 2000. View at Publisher · View at Google Scholar
  32. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Physical Review E, vol. 71, no. 5, Article ID 056619, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  33. Z. Y. Yan, “Nonautonomous “rogons” in the inhomogeneous nonlinear Schrödinger equation with variable coefficients,” Physics Letters A, vol. 374, no. 4, pp. 672–679, 2010. View at Publisher · View at Google Scholar
  34. F. C. You, T. C. Xia, and J. Zhang, “Frobenius integrable decompositions for two classes of nonlinear evolution equations with variable coefficients,” Modern Physics Letters B, vol. 23, no. 12, pp. 1519–1524, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. X.-Y. Tang, Y. Gao, F. Huang, and S.-Y. Lou, “Variable coefficient nonlinear systems derived from an atmospheric dynamical system,” Chinese Physics B, vol. 18, no. 11, pp. 4622–4635, 2009. View at Publisher · View at Google Scholar · View at Scopus
  36. X. Y. Yu, Y.-T. Gao, Z.-Y. Sun, and Y. Liu, “N-soliton solutions, Bäcklund transformation and Lax pair for a generalized variable-coefficient fifth-order Korteweg-de Vries equation,” Physica Scripta, vol. 81, no. 4, Article ID 045402, 2010. View at Publisher · View at Google Scholar · View at Scopus
  37. J. Weiss, M. Tabor, and G. Carnevale, “The Painlevé property for partial differential equations,” Journal of Mathematical Physics, vol. 24, no. 3, pp. 522–526, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. B. Chen and Y. C. Xie, “An auto-Bäcklund transformation and exact solutions of stochastic Wick-type Sawada-Kotera equations,” Chaos, Solitons & Fractals, vol. 23, no. 1, pp. 243–248, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet