Advertisement

Theoretical and Mathematical Physics

, Volume 196, Issue 3, pp 1320–1332 | Cite as

Generalized Darboux Transformation for the Discrete Kadomtsev–Petviashvili Equation with Self-Consistent Sources

  • Runliang Lin
  • Yukun Du
Article
  • 13 Downloads

Abstract

We construct several types of Darboux transformations for the discrete Kadomtsev–Petviashvili equation with self-consistent sources (dKPwS) including the elementary Darboux transformation, the adjoint Darboux transformation, and the binary Darboux transformation. These Darboux transformations can be used to obtain some solutions of the dKPwS. We give some solutions explicitly.

Keywords

integrable system with self-consistent sources discrete Kadomtsev–Petviashvili equation Darboux transformation Hirota equation soliton solution 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM Stud. Appl. Math., Vol. 4), SIAM, Philadelphia (1981).CrossRefzbMATHGoogle Scholar
  2. 2.
    E. V. Doktorov and V. S. Shchesnovich, “Nonlinear evolutions with singular dispersion laws associated with a quadratic bundle,” Phys. Lett. A, 207, 153–158 (1995).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    P. G. Grinevich and I. A. Taimanov, “Spectral conservation laws for periodic nonlinear equations of the Melnikov type,” in: Geometry, Topology, and Mathematical Physics:S. P. Novikov’s Seminar: 2006–2007 (AMS Transl. Ser. 2, Vol. 224, V. M. Buchstaber and I. M. Krichever, eds.), Amer. Math. Soc., Providence, R. I. (2008), pp. 125–138; arXiv:0801.4143v1 [math-ph] (2008).Google Scholar
  4. 4.
    A. M. Kamchatnov and M. V. Pavlov, “Periodic waves in the theory of self-induced transparency,” JETP, 80, 22–27 (1995).ADSGoogle Scholar
  5. 5.
    V. K. Mel’nikov, “On equations for wave interactions,” Lett. Math. Phys., 7, 129–136 (1983).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    R. L. Lin, Y. B. Zeng, and W.-X. Ma, “Solving the KdV hierarchy with self-consistent sources by inverse scattering method,” Phys. A, 291, 287–298 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    X.-B. Hu and H.-Y. Wang, “Construction of dKP and BKP equations with self-consistent sources,” Inverse Probl., 22, 1903–1920 (2006).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    D.-J. Zhang and D.-Y. Chen, “The N-soliton solutions of the sine-Gordon equation with self-consistent sources,” Phys. A, 321, 467–481 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    R. L. Lin, H. S. Yao, and Y. B. Zeng, “Restricted flows and the soliton equation with self-consistent sources,” SIGMA, 2, 096 (2006); arXiv:nlin.SI/0701003v1 (2007).MathSciNetzbMATHGoogle Scholar
  10. 10.
    Y. Zeng, Y. Shao, and W. Xue, “Negaton and positon solutions of the soliton equation with self-consistent sources,” J. Phys. A: Math. Gen., 36, 5035–5043 (2003).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    R. L. Lin, H. Peng, and M. Ma˜nas, “The q-deformed mKP hierarchy with self-consistent sources, Wronskian solutions, and solitons,” J. Phys. A: Math. Theor., 43, 434022 (2010).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    X. J. Liu, R. L. Lin, B. Jin, and Y. B. Zeng, “A generalized dressing approach for solving the extended KP and the extended mKP hierarchy,” J. Math. Phys., 50, 053506 (2009).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    R. L. Lin, T. C. Cao, X. J. Liu, and Y. B. Zeng, “Bilinear identities for an extended B-type Kadomtsev–Petviashvili hierarchy,” Theor. Math. Phys., 186, 307–319 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    R. L. Lin, X. J. Liu, and Y. B. Zeng, “Bilinear identities and Hirota’s bilinear forms for an extended Kadomtsev–Petviashvili hierarchy,” J. Nonlinear Math. Phys., 20, 214–228 (2013).MathSciNetCrossRefGoogle Scholar
  15. 15.
    X. J. Liu, Y. B. Zeng, and R. L. Lin, “A new extended KP hierarchy,” Phys. Lett. A, 372, 3819–3823 (2008).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    X.-P. Cheng, C.-L. Chen, and S. Y. Lou, “Interactions among different types of nonlinear waves described by the Kadomtsev–Petviashvili equation,” Wave Motion, 51, 1298–1308 (2014).MathSciNetCrossRefGoogle Scholar
  17. 17.
    A. Doliwa and R. L. Lin, “Discrete KP equation with self-consistent sources,” Phys. Lett. A, 378, 1925–1931 (2014).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    O. Chvartatskyi, A. Dimakis, and F. Müller-Hoissen, “Self-consistent sources for integrable equations via deformations of binary Darboux transformations,” Lett. Math. Phys., 106, 1139–1179 (2016).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    A. N. W. Hone, T. E. Kouloukas, and C. Ward, “On reductions of the Hirota–Miwa equation,” SIGMA, 13, 057 (2017).MathSciNetzbMATHGoogle Scholar
  20. 20.
    A. K. Pogrebkov, “Symmetries of the Hirota difference equation,” SIGMA, 13, 053 (2017).MathSciNetzbMATHGoogle Scholar
  21. 21.
    A. V. Zabrodin, “Hirota’s difference equations,” Theor. Math. Phys., 113, 1347–1392 (1997).MathSciNetCrossRefGoogle Scholar
  22. 22.
    R. Hirota, The Direct Method in Soliton Theory (Cambridge Tracts Math., Vol. 155), Cambridge Univ. Press, Cambridge (2004).CrossRefzbMATHGoogle Scholar
  23. 23.
    J. J. C. Nimmo, “Darboux transformations and the discrete KP equation,” J. Phys. A: Math. Gen., 30, 8693–8704 (1997).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    J. J. C. Nimmo, “On a non-Abelian Hirota–Miwa equation,” J. Phys. A: Math. Gen., 39, 5053–5065 (2006).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    T. Miwa, “On Hirota’s difference equations,” Proc. Japan Acad. Ser. A Math. Sci., 58, 9–12 (1982).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    R. M. Kashaev, “On discrete three-dimensional equations associated with the local Yang–Baxter relation,” Lett. Math. Phys., 38, 389–397 (1996).ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Department of Mathematical Sciences, School of SciencesTsinghua UniversityBeijingChina

Personalised recommendations