Symbolic algorithms for the Painlevé test, special solutions, and recursion operators for nonlinear PDEs. This paper discusses the algorithms and implementations of three MATHEMATICA packages for the study of integrability and the computation of closed-form solutions of nonlinear polynomial partial differential equations (PDEs). The first package, PainleveTest.m, symbolically performs the Painlevé integrability test. The second package, PDESpecialSolutions.m, computes exact solutions expressible in hyperbolic or elliptic functions. The third package, PDERecursionOperator.m, generates and tests recursion operators.

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  1. Wazwaz, Abdul-Majid: Negative-order integrable modified KdV equations of higher orders (2018)
  2. Wazwaz, Abdul-Majid: Painlevé analysis for a new integrable equation combining the modified Calogero-Bogoyavlenskii-Schiff (MCBS) equation with its negative-order form (2018)
  3. Özemir, C.: On some canonical classes of cubic-quintic nonlinear Schrödinger equations (2017)
  4. Gómez-Ullate, D.; Santini, P. M.; Sommacal, M.; Calogero, F.: Understanding complex dynamics by means of an associated Riemann surface (2012)
  5. Baldwin, D. E.; Hereman, W.: A symbolic algorithm for computing recursion operators of nonlinear partial differential equations (2010)
  6. Xu, Gui-Qiong: A note on the Painlevé test for nonlinear variable-coefficient PDEs (2009)
  7. Zhao, Yin-Long; Liu, Yin-Ping; Li, Zhi-Bin: A modified WTC algorithm for the Painlevé test of nonlinear variable-coefficient PDEs (2009)
  8. Picard, P. Y.: Some exact solutions of the ideal MHD equations through symmetry reduction method (2008)
  9. Picard, P. Y.: Some spherical solutions of ideal magnetohydrodynamic equations (2007)
  10. Baldwin, Douglas; Hereman, Willy: Symbolic software for the Painlevé test of nonlinear ordinary and partial differential equations (2006)
  11. Vernov, S. Yu.: Construction of exact partial solutions of nonintegrable systems by means of formal Laurent and Puiseux series (2006)
  12. Baldwin, D.; Hereman, W.; Sayers, J.: Symbolic algorithms for the Painlevé test, special solutions, and recursion operators for nonlinear PDEs (2005)