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Integrable discretization of soliton equations via bilinear method and Bäcklund transformation

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Abstract

We present a systematic procedure to derive discrete analogues of integrable PDEs via Hirota’s bilinear method. This approach is mainly based on the compatibility between an integrable system and its Bäcklund transformation. We apply this procedure to several equations, including the extended Korteweg-de-Vries (KdV) equation, the extended Kadomtsev-Petviashvili (KP) equation, the extended Boussinesq equation, the extended Sawada-Kotera (SK) equation and the extended Ito equation, and obtain their associated semi-discrete analogues. In the continuum limit, these differential-difference systems converge to their corresponding smooth equations. For these new integrable systems, their Bäcklund transformations and Lax pairs are derived.

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References

  1. Ablowitz M J, Ladik J F. Nonlinear differential-difference equations and Fourier analysis. J Math Phys, 1975, 16: 598–605

    Article  MATH  MathSciNet  Google Scholar 

  2. Chiu S C, Ladik J F. Generating exactly soluble nonlinear discrete evolution equations by a generalized Wronskian technique. J Math Phys, 1977, 18: 690–700

    Article  MATH  MathSciNet  Google Scholar 

  3. Date E, Jimbo M, Miwa T. Methods for generating discrete soliton equations I. J Phys Soc Japan, 1982, 51: 4116–4124

    Article  MathSciNet  Google Scholar 

  4. Date E, Jimbo M, Miwa T. Methods for generating discrete soliton equations II. J Phys Soc Japan, 1982, 51: 4125–4131

    Article  Google Scholar 

  5. Date E, Jimbo M, Miwa T. Methods for generating discrete soliton equations III. J Phys Soc Japan, 1983, 52: 388–393

    Article  MATH  MathSciNet  Google Scholar 

  6. Date E, Jimbo M, Miwa T. Methods for generating discrete soliton equations IV. J Phys Soc Japan, 1983, 52: 761–765

    Article  MATH  MathSciNet  Google Scholar 

  7. Date E, Jimbo M, Miwa T. Methods for generating discrete soliton equations V. J Phys Soc Japan, 1983, 52: 766–771

    Article  Google Scholar 

  8. Faddeev L, Volkov A Y. Hirota equation as an example of integrable symplectic map. Lett Math Phys, 1994, 32: 125–135

    Article  MATH  MathSciNet  Google Scholar 

  9. Feng B F, Maruno K, Ohta Y. Integrable discretizations of the short pulse equation. J Phys A Math Gen, 2010, 43: 085203

    Article  MathSciNet  Google Scholar 

  10. Feng B F, Maruno K, Ohta Y. Integrable discretizations for the short-wave model of the Camassa-Holm equation. J Phys A Math Gen, 2010, 43: 265202

    Article  MathSciNet  Google Scholar 

  11. Grammaticos B, Kosmann-Schwarzbach Y, Tamizhmani T. Discrete Integrable Systems. Berlin: Springer, 2004

    Book  MATH  Google Scholar 

  12. Hirota R. A new form of Bäcklund transformations and its relation to the inverse scattering problem. Prog Theor Phys, 1974, 52: 1498–1512

    Article  MATH  Google Scholar 

  13. Hirota R. Nonlinear partial difference equations I. A difference analogue of the Korteweg-de Vries equation. J Phys Soc Japan, 1977, 43: 1424–1433

    Article  MathSciNet  Google Scholar 

  14. Hirota R. Nonlinear partial difference equations, II: Discrete-time Toda equation. J Phys Soc Japan, 1977, 43: 2074–2078

    Article  MathSciNet  Google Scholar 

  15. Hirota R. Nonlinear partial difference equations, III: Discrete Sine-Gordon equation. J Phys Soc Japan, 1977, 43: 2079–2086

    Article  MathSciNet  Google Scholar 

  16. Hirota R. Direct Method in Soliton Theory. Cambridge: Cambridge University Press, 2004

    Book  MATH  Google Scholar 

  17. Hu X B. Hirota-type equation, soliton solutions, Bäcklund transformations and conservation laws. J Partial Differential Equations, 1990, 3: 87–95

    MATH  MathSciNet  Google Scholar 

  18. Hu X B, Clarkson P A. Bäcklund transformations and nonlinear superposition formulae of a differential-difference KdV equation. J Phys A Math Gen, 1998, 31: 1405–1414

    Article  MATH  MathSciNet  Google Scholar 

  19. Hu X B, Li Y. Nonlinear superposition formulae of the Ito equation and a model equation for shallow water waves. J Phys A Math Gen, 1991, 24: 1979–1986

    Article  MATH  Google Scholar 

  20. Hu X B, Yu G F. Integrable semi-discretizations and full-discretization of the two-dimensional Leznov lattice. J Differ Equ Appl, 2009, 15: 233–252

    Article  MATH  MathSciNet  Google Scholar 

  21. Hu X B, Zhu Z N, Wang D L. A differential-difference Caudrey-Dodd-Gibbon-Kotera-Sawada equation. J Phys Soc Japan, 2000, 69: 1042–1049

    Article  MATH  MathSciNet  Google Scholar 

  22. Huang X C. A two-parameter Bäcklund transformation for the Boussinesq equation. J Phys A Math Gen, 1982, 15: 3367–3372

    Article  MATH  Google Scholar 

  23. Ito M. An Extension of Nonlinear Evolution Equations of the K-dV (mK-dV) Type to Higher Orders. J Phys Soc Japan, 1980, 49: 771–778

    Article  MathSciNet  Google Scholar 

  24. Kakei S. Introduction of discrete integrable systems (in Japanese). MI Lect Note Ser, 2012, 40: 27–49

    Google Scholar 

  25. Levi D. Nonlinear differential difference equations as Bäcklund transformations. J Phys A Math Gen, 1981, 14: 1083–1098

    Article  MATH  Google Scholar 

  26. Levi D, Benguria R. Bäcklund transformations and nonlinear differential difference equations. Proc Natl Acad Sci USA, 1980, 77: 5025–5027

    Article  MATH  MathSciNet  Google Scholar 

  27. Levi D, Pilloni L, Santini P M. Integrable three-dimensional lattices. J Phys A Math Gen, 1981, 14: 1567–1575

    Article  MATH  MathSciNet  Google Scholar 

  28. Levi D, Ragnisco O. SIDE III-Symmetries and Integrability of Difference Equations. Montreal: Amer Math Soc, 2000

    MATH  Google Scholar 

  29. Nakamura A. Decay mode solution of the two-dimensional KdV equation and the generalized Bäcklund transformation. J Math Phys, 1981, 22: 2456–2462

    Article  MATH  MathSciNet  Google Scholar 

  30. Nijhoff F W, Capel H W, Wiersma G L, et al. Linearizing integral transform and partial difference equations. Phys Lett A, 1984, 103: 293–297

    Article  MathSciNet  Google Scholar 

  31. Ohta Y, Hirota R. A discrete KdV equation and its Casorati determinant solution. J Phys Soc Japan, 1991, 60: 2095–2095

    Article  MathSciNet  Google Scholar 

  32. Ohta Y, Maruno K, Feng B F. An integrable semi-discretization of the Camassa-Holm equation and its determinant solution. J Phys A Math Gen, 2008, 41: 355205

    Article  MathSciNet  Google Scholar 

  33. Petrera M, Suris Y B. On the Hamiltonian structure of Hirota-Kimura discretization of the Euler top. Math Nachr, 2010, 283: 1654–1663

    Article  MATH  MathSciNet  Google Scholar 

  34. Petrera M, Suris Y B. SV Kovalevskaya system, its generalization and discretization. Front Math China, 2013, 8: 1047–1065

    Article  MATH  MathSciNet  Google Scholar 

  35. Quispel G R W, Nijhoff F W, Capel H W, et al. Linear integral equations and nonlinear differrencedifference equations. Phys A, 1984, 125: 344–380

    Article  MATH  MathSciNet  Google Scholar 

  36. Schiff J. Loop groups and discrete KdV equations. Nonlinearity, 2003, 16: 257–275

    Article  MATH  MathSciNet  Google Scholar 

  37. Suris Y B. The problem of integrable discretization: Hamiltonian approach. Boston, MA: Birkhäuser, 2003

    Book  Google Scholar 

  38. Veni S S, Latha M M. A generalized Davydov model with interspine coupling and its integrable discretization. Phys Scr, 2012, 86: 025003

    Article  Google Scholar 

  39. Vinet L, Yu G F. On the discretization of the coupled integrable dispersionless equations. Journal of Nonlinear Math Phys, 2013, 20: 106–125

    Article  MathSciNet  Google Scholar 

  40. Vinet L, Yu G F. Discrete analogues of the generalized coupled integrable dispersionless equations. J Phys A Math Gen, 2013, 46: 175205–175220

    Article  MathSciNet  Google Scholar 

  41. Zhang Y, Hu X B, Tam H W. Integrable discretization of nonlinear Schrödinger equation and its application with Fourier pseudo-spectral method. Numer Algorithms, 2014, doi: 10.1007/s11075-014-9928-7

    Google Scholar 

  42. Zhang Y, Tam H W, Hu X B. Integrable discretization of time and its application on the Fourier pseudospectral method to the Kortewegde Vries equation. J Phys A Math Gen, 2014, 47: 045202

    Article  MathSciNet  Google Scholar 

  43. Zhou T, Zhu Z N, He P. A fifth order semidiscrete mKdV equation. Sci China Math, 2013, 56: 123–134

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Computational Mathematics and Scientific Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    YingNan Zhang, XiangKe Chang & XingBiao Hu

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    YingNan Zhang & XiangKe Chang

  3. Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou, 310023, China

    Juan Hu

  4. Department of Computer Science, Hong Kong Baptist University, Hong Kong, China

    Hon-Wah Tam

Authors
  1. YingNan Zhang
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  2. XiangKe Chang
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  3. Juan Hu
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  4. XingBiao Hu
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  5. Hon-Wah Tam
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Correspondence to YingNan Zhang.

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Zhang, Y., Chang, X., Hu, J. et al. Integrable discretization of soliton equations via bilinear method and Bäcklund transformation. Sci. China Math. 58, 279–296 (2015). https://doi.org/10.1007/s11425-014-4952-6

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  • DOI: https://doi.org/10.1007/s11425-014-4952-6

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