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Multi-linear variable separation approach to nonlinear systems

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Abstract

The multi-linear variable separation approach is reviewed in this article. The method has been recently established and successfully solved a large number of nonlinear systems. One of the most exciting findings is that the basic multi-linear variable separation solution can be expressed by a universal formula including two (1+1)-dimensional functions, and at least one is arbitrary for integrable systems. Furthermore, the method has been extended in two different ways so as to enroll more low dimensional functions in the solution.

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References

  1. C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Phys. Rev. Lett., 1967, 19: 1095

    Article  MATH  ADS  Google Scholar 

  2. M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge: Cambridge University Press, 1991

    MATH  Google Scholar 

  3. A. Boutet de Monvel, A. S. Fokas, and D. Shepelsky, Lett. Math. Phys., 2003, 65: 199

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. A. S. Fokas, Nonlinearity, 2004, 17: 1521

    Article  MATH  MathSciNet  Google Scholar 

  5. A. S. Fokas and A. R. Its, J. Phys. A: Math Gen., 2004, 37: 6091

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. A. Boutet de Monvel, A. S. Fokas, and D. Shepelsky, Comm. Math. Phys., 2006, 263: 133

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. C. W. Cao, Sci. China Ser. A, 1990, 33: 528

    MATH  MathSciNet  Google Scholar 

  8. Y. Cheng and Y. S. Li, Phys. Lett. A, 1991, 175: 22

    Article  ADS  MathSciNet  Google Scholar 

  9. S. Y. Lou and L. L. Chen, J. Math. Phys., 1999, 40: 6491

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. K. S. Chou and C. Z. Qu, J. Phys. A: Math. Gen., 1999, 32: 6271

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. C. Z. Qu, S. L. Zhang, and R. C. Liu, Physica D, 2000, 144: 97

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. P. G. Estevez, C. Z. Qu, and S. L. Zhang, J. Math. Anal. Appl., 2002, 275: 44

    Article  MATH  MathSciNet  Google Scholar 

  13. S. L. Zhang, S. Y. Lou, and C. Z. Qu, J. Phys. A: Math. Gen., 2003, 36: 12223

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. S. L. Zhang and S. Y. Lou, Physica A, 2004, 335: 430

    Article  ADS  MathSciNet  Google Scholar 

  15. S. Y. Lou and J. Z. Lu, J. Phys. A: Math. Gen., 1996, 29: 4209

    Article  MATH  ADS  MathSciNet  Google Scholar 

  16. X. Y. Tang, S. Y. Lou, and Y. Zhang, Phys. Rev. E, 2002, 66: 046601

    Article  ADS  MathSciNet  Google Scholar 

  17. X. Y. Tang and S. Y. Lou, Commun. Theor. Phys., 2002, 38: 1

    MathSciNet  Google Scholar 

  18. X. Y. Tang, J. M. Li, and S. Y. Lou, Phys. Scr., 2007, 75: 201

    Article  ADS  MathSciNet  Google Scholar 

  19. S. Y. Lou and X. Y. Tang, Nonlinear Mathematical Physics Methods, Bei**g: Science Press, 2006

    Google Scholar 

  20. X. Y. Tang, C. L. Chen, and S. Y. Lou, J. Phys. A: Math. Gen., 2002, 35: L293

    Article  MATH  ADS  MathSciNet  Google Scholar 

  21. X. Y. Tang, K. W. Chow, and S. Y. Lou, J. Phys. A: Math. Gen., 2005, 38: 10361

    Article  MATH  ADS  MathSciNet  Google Scholar 

  22. X. Y. Tang, Phys. Lett. A, 2003, 314: 286

    Article  MATH  ADS  MathSciNet  Google Scholar 

  23. S. Y. Lou, C. L. Chen, and X. Y. Tang, J. Math. Phys., 2002, 43: 4078

    Article  MATH  ADS  MathSciNet  Google Scholar 

  24. A. Maccaria, Phys. Lett. A, 2005, 336: 117

    Article  ADS  MathSciNet  Google Scholar 

  25. A. Maccaria, Chaos, Solitons and Fractals, 2006, 27: 363

    Article  Google Scholar 

  26. A. Maccaria, J. Math. Phys., 2008, 49: 022702

    Article  ADS  MathSciNet  Google Scholar 

  27. B. B. Thomas, K. K. Victor, and K. T. Crepin, J. Phys. A: Math. Theor., 2008, 41: 135208

    Article  ADS  Google Scholar 

  28. A. M. Wazwaz, Appl. Math. Comput., 2008, 204: 817

    Article  MATH  MathSciNet  Google Scholar 

  29. J. F. Zhang, C. Q. Dai, C. Z. Xu, J. P. Meng, and X. J. Lai, Phys. Lett. A, 2006, 352: 511

    Article  ADS  MathSciNet  Google Scholar 

  30. J. P. Ying and S. Y. Lou, Chin. Phys. Lett., 2003, 20: 1448

    Article  ADS  Google Scholar 

  31. X. Y. Tang and Z. F. Liang, Phys. Lett. A, 2006, 351: 398

    Article  ADS  MathSciNet  Google Scholar 

  32. X. M. Qian, S. Y. Lou, and X. B. Hu, J. Phys. A: Gen. Math., 2004, 37: 2401

    Article  MATH  ADS  MathSciNet  Google Scholar 

  33. X. M. Qian, S. Y. Lou, and X. B. Hu, Z. Naturforsch., 2004, 59a: 645

    Google Scholar 

  34. S. F. Shen, Phys. Lett. A, 2007, 365: 210

    Article  ADS  MathSciNet  Google Scholar 

  35. X. Y. Tang and S. Y. Lou, J. Math. Phys., 2003, 44: 4000

    Article  MATH  ADS  MathSciNet  Google Scholar 

  36. X. Y. Tang and S. Y. Lou, Commun. Theor. Phys., 2003, 40: 62

    MathSciNet  Google Scholar 

  37. X. Y. Tang and S. Y. Lou, Chin. Phys. Lett., 2003, 3: 335

    ADS  Google Scholar 

  38. W. K. Schief, Proc. R. Soc. London Ser. A, 1997, 453: 1671

    Article  MATH  ADS  MathSciNet  Google Scholar 

  39. R. Hirota, Phys. Rev. Lett., 1971, 27: 1192

    Article  MATH  ADS  Google Scholar 

  40. H. C. Hu, S. Y. Lou, and Q. P. Liu, Chin. Phys. Lett., 2003, 20: 1413

    Article  ADS  Google Scholar 

  41. H. C. Hu, X. Y. Tang, S. Y. Lou, and Q. P. Liu, Chaos, Soltions and Fractals, 2004, 22: 327

    Article  MATH  MathSciNet  Google Scholar 

  42. H. C. Hu and S. Y. Lou, Chin. Phys. Lett., 2004, 21: 2073

    Article  ADS  Google Scholar 

  43. C. Q. Dai, C. J. Yan, and J. F. Zhang, Commun. Theor. Phys., 2006, 46: 389

    Article  MathSciNet  Google Scholar 

  44. C. Q. Dai and J. F. Zhang, J. Math. Phys., 2006, 47: 043501

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Physics, Shanghai Jiao Tong University, Shanghai, 200240, China

    **ao-yan Tang  (唐晓艳) & Sen-yue Lou  (楼森岳)

  2. Department of Physics, Ningbo University, Ningbo, 315211, China

    Sen-yue Lou  (楼森岳)

  3. School of Mathematics, Fudan University, Shanghai, 200433, China

    Sen-yue Lou  (楼森岳)

Authors
  1. **ao-yan Tang  (唐晓艳)
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  2. Sen-yue Lou  (楼森岳)
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Correspondence to Sen-yue Lou  (楼森岳).

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Tang, Xy., Lou, Sy. Multi-linear variable separation approach to nonlinear systems. Front. Phys. China 4, 235–240 (2009). https://doi.org/10.1007/s11467-009-0046-2

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  • DOI: https://doi.org/10.1007/s11467-009-0046-2

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