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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Partially Integrable &weierp T -Symmetric Hierarchies of the KdV and Burgers' Equations in (1+1) and (2+1)

Discontinuity, Nonlinearity, and Complexity 6(2) (2017) 113--146 | DOI:10.5890/DNC.2017.06.002

Keri Pecora; S. Roy Choudhury

Department of Mathematics, University of Central Florida, Orlando, FL 32816-1364 USA

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In this paper, we generalize the work of Bender and co-workers to derive new partially-integrable hierarchies of various &weierp T -symmetric, nonlinear partial differential equations. The possible integrable members are identified employing the Painlevé Test, a necessary but not sufficient, integrability condition, and are indexed by the integer n, corresponding to the negative of the order of the dominant pole in the singular part of the Painlevé expansion for the solution. For the &weierp T -symmetric Korteweg-de Vries (KdV) equation, as with some other hierarchies, the first or n = 1 equation fails the test, the n = 2 member corresponds to the regular KdV equation, while the remainder form an entirely new, possibly integrable, hierarchy. Bäcklund Transformations and analytic solutions of the n = 3 and n = 4 members are derived. The solutions, or solitary waves, prove to be algebraic in form. The &weierp T -symmetric Burgers’ equation fails the Painlevé Test for its n = 2 case, but special solutions are nonetheless obtained. Also, a &weierp T - Symmetric hierarchy of the (2+1) Burgers’ equation is analyzed. The Painlevé Test and invariant Painlevé analysis in (2+1) dimensions are utilized, and BTs and special solutions are found for those cases that pass the Painlevé Test.


  1. [1]  Ramani, A., Grammaticos, B., and Bountis, T. (1989), The Painlevé property and singularity sanalysis of integrable and non-integrable systems, Phys. Rep., 180, 160.
  2. [2]  Ablowitz, M.J., Ramani, A., and Segur, H. (1980), A connection between nonlinear evolution equations and ODEs of P-type: I and II, J. Math. Phys., 21, 715-1006.
  3. [3]  Weiss, J., Tabor,M., and Carnevale, G. (1983), The Painlevé property for partial differential equations, J. Math. Phys., 24, 522.
  4. [4]  Hietarinta, J. (1987), Direct methods for the search of the second invariant, Phys. Rep., 147, 87.
  5. [5]  Ablowitz, M.J. and Segur, H. (1981), Solitons and the Inverse Scattering Transform, SIAM, Philadelphia; Dodd, R.K., Eilbec, J.C., Gibbon, J.D., and Morris, H.C. (1982), Solitons and Nonlinear Wave Equations, Academic Press, New York.
  6. [6]  Tabor, M. (1989), Chaos and integrability in nonlinear dynamics: an introduction,Wiley, New York.
  7. [7]  Ince, E.L. (1956), Ordinary Differential Equations, Dover, New York.
  8. [8]  Weiss, J. (1983), (1984), (1985), (1986), (1987), The Painlevé property for partial differential equations. II, J. Math. Phys., 24, 1405; 25(13), 2226; 26(258), 2174; 27, 1293-2647; 28, 2025.
  9. [9]  Kudryashov, N.A. (1990), Exact solution of the generalized KS equation, Phys. Lett. A, 147, 287.
  10. [10]  Conte, R. and Musette, M. (1989), Painlevé analysis and BT in the Duramoto-Sivashinsky equation, J. Phys. A, 22, 169.
  11. [11]  Cariello, F. and Tabor, M. (1989), Painlevé expansions for nonintegrable evolution equations, Physica D, 39, 77.
  12. [12]  Roy Choudhury, S. (1991), Painlevé analysis and special solutions of two families of reaction-diffusion equations, Phys. Lett. A, 159, 311.
  13. [13]  Yu Guo, B. and Xiong, Z. (1991), Analytic solutions of the Fisher equation, J. Phys. A, 24, 645.
  14. [14]  Roy Choudhury, S. and BTs, (1992), truncated Painlevé expansions and special solutions of nonintegrable long-wave evolution equations, Can. J. Phys., 70, 595; (1992), Painlevé analysis and partial integrability of a class of reactiondiffusion equations, Nonlin. Anal: Theory, Meth. & Appl., 18, 445.
  15. [15]  Newell, A.C., Tabor M., and Zeng, Y.B. (1987), A unified approach to Painlevé expansions, Physica D, 29, 1.
  16. [16]  Flaschka, H., Newell, A.C., and Tabor, M. (1991), Monodromy- and spectrum-preserving deformations, in What is Integrability, V.E. Zakharov (Ed.), Springer, Berlin.
  17. [17]  Hille, E. (1976), ODEs in the Complex Domain,Wiley, New York.
  18. [18]  Conte, R. (1989), Invariant Painlevé analysis of PDEs, Phys. Lett. A, 140, 383.
  19. [19]  Musette, M. and Conte, R. (1991), Algorithmic method for deriving Lax pairs from the invariant Painlevé analysis of NLPDEs, J. Math. Phys., 32, 1450.
  20. [20]  Musette, M. and Conte, R. (1994), The two-singular-manifold method: I. MKdV and sine-Gordon equations, J. Phys. A: Math. Gen, 27, 3895.
  21. [21]  Roy Choudhury, S. (1999), Invariant Painlevé analysis and coherent structures of two families of reaction-diffusion equations, J. Math. Phys., 40, 3643.
  22. [22]  Roy Choudhury, S. (2000), One and 2D coherent structures of the Zakharov-Kuznetsov equations, Problems of Nonlin. Anal., 6, 1.
  23. [23]  Isldore, N. and Malfliet, W. (1997), New special solutions of the ‘Brusselator’ reaction model, J. Phys. A: Math. Gen, 30, 5151.
  24. [24]  Conte, R. and Musette, M. (1993), Linearity inside nonlinearity: Exact solutions of the 1D Quintic CGL equation, Physica D, 69, 1.
  25. [25]  Estevez, P.G., Conde E., and Gordoa, P.R. (1998), Unified approach to Miura, Bäcklund and Darboux transformations for NLPDEs, J. Math. Phys., 5, 82.
  26. [26]  Estevez, P.G. and Gordoa, P.R. (1997), Darboux transformations via Painleé analysis, Inverse Problems, 13, 939.
  27. [27]  Wadati, M., Sanuki, H., and Konno, K. (1975), Relationships among the inverse method, BTs, and an infinite number of conservation laws, Prog. Theor. Phys., 53, 419.
  28. [28]  Fring, A. (2007), PT -symmetric deformations of the Korteweg-de-Vries equation, J. Phys. A: Math. Theor., 40, 4215-4224; Bender, C.M., Brody, D.C., Chen J., and Furlan, E. (2006)PT -symmetric Extension of the Korteweg-de Vries Equation, arXiv:math-ph/0610003v1.
  29. [29]  Yan, Z. (2008), Complex PT -symmetric extensions of the non-PT -symmetric Burgers equation, Physica Scripta., 77, 025006.
  30. [30]  Bender, C.M. (2007),Making Sense of non-Hermitian Hamiltonians, Reports on Progress in Physics, 70, 947-1018.
  31. [31]  En-Gui, F., Hong-Qing, Z., and Gang, L. (1998), Bäcklund Transformation, Lax Pairs, Symmetries and Exact Solutions for Variable Coefficient KdV Equation, Acta Phys. Sin. (Overseas Edn.), 7, 649.
  32. [32]  Roy Choudhury, S. (2002), A Unified Approach to Integrable Systems via Painlevé Analysis,Contemporary Mathematics, 301, 139. pg 322-357.
  33. [33]  Roy Choudhury, S. (2000), Invariant Painlevé Analysis and Coherent Structures of Long-wave Equations, Physica Scripta., 62, 156.
  34. [34]  Tanriver U. and Roy Choudhury, S. (2000), One and Two Dimensional Coherent Structures of the Zakharov-Kuznetsov Equation via Invariant Painlevé Analysis, Problems of Nonlinear Analysis of Engineering Systems, 6.
  35. [35]  Stephani, H. (1989), Differential Equations - Their Solutions Using Symmetries, edM. MacCullum, CambridgeUniv. Press, Cambridge.
  36. [36]  Hydon, P.E. (1999), Symmetry Methods for Differential Equations, Cambridge Univ. Press, Cambridge.
  37. [37]  Fels, M.E. (1996), The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations, Trans. Amer. Math. Soc., 348, 5007-5029.
  38. [38]  Nucci, M.C. and Arthurs, A.M. (2010), On the inverse problem of calculus for fourth-order equations, Proc. R. Soc. A, 466, doi: 10.1098/rspa.2009.0618.
  39. [39]  Kaup, D. J. and Malomed, B. (2003), Embedded Solitons in Lagrangan and Semi-Lagrangian Systems, Physica D, 184, 153-161.
  40. [40]  Smith, T. B. and Roy Choudhury, S. (2009), Regular and Embedded Solitons in a Generalized Pochammer PDE, Comm. Nonlin. Sci. Numer. Simulation, 14, 2637-2641.