A symbolic algorithm for computing recursion operators of nonlinear partial differential equations

References

• Ablowitz, M. J. , and Clarkson, P. A. , 1991. Solitons, Nonlinear Evolution Equations and Inverse Scattering . London Mathematical Society Lecture Note Series Vol. 149. Cambridge: Cambridge University Press; 1991.
   Google Scholar
• Ablowitz, M. J. , and Segur, H. , 1981. Solitons and the Inverse Scattering Transform . Philadelphia, PA: SIAM; 1981, IAM Studies in Applied Mathematics.
   Google Scholar
• Ablowitz, M. J. , Ramani, A. , and Segur, H. , 1980. A connection between nonlinear evolution equations and ordinary differential equations of P-type. I , J. Math. Phys. 21 (1980), pp. 715–721.
   Web of Science ®Google Scholar
• Adams, P. J. , 2003. "Symbolic computation of conserved densities and fluxes for systems of partial differential equations with transcendental nonlinearities". Golden, Colorado: Colorado School of Mines; 2003, M.S. thesis.
   Google Scholar
• P.J. Adams and W. Hereman, TransPDEDensityFlux.m: A Mathematica package for the symbolic computation of conserved densities and fluxes for systems of partial differential equations with transcendental nonlinearities, 2002; software available at http://inside.mines.edu/~whereman/software/TransPDEDensityFlux.
   Google Scholar
• Baldwin, D. , 2004. "Symbolic algorithms and software for the Painlevé test and recursion operators for nonlinear partial differential equations". Golden, Colorado: Colorado School of Mines; 2004, M.S. thesis.
   Google Scholar
• D. Baldwin and W. Hereman, PainleveTest.m: A Mathematica package for the Painlevé test of systems of ODEs and PDEs, 2003; software available at http://inside.mines.edu/~whereman/software/painleve/mathematica.
   Google Scholar
• Baldwin, D. , and Hereman, W. , 2006. Symbolic software for the Painlevé test of nonlinear ordinary and partial differential equations , J. Nonlinear Math. Phys. 13 (2006), pp. 90–110.
   Web of Science ®Google Scholar
• D. Baldwin and W. Hereman, PDERecursionOperator.m: A Mathematica package for the symbolic computation of recursion operators for nonlinear partial differential equations, 2003–2009; software available at http://inside.mines.edu/~whereman/software/PDERecursionOperator.
   Google Scholar
• Baldwin, D. , Hereman, W. , and Sayers, J. , 2005. "Symbolic algorithms for the Painlevé test, special solutions, and recursion operators for nonlinear PDEs". In: Winternitz, P. , Gomez-Ullate, D. , Iserles, A. , Levi, D. , Olver, P. J. , Quispel, R. , and Tempesta, P. , eds. Group Theory and Numerical Analysis . CRM Proc. & Lect. Ser. Vol. 39. Providence, RI: AMS; 2005. pp. 17–32.
   Google Scholar
• Bilge, A. H. , 1993. On the equivalence of linearization and formal symmetries as integrability tests for evolution equations , J. Phys. A. Math. Gen. 26 (1993), pp. 7511–7519.
   Google Scholar
• Calogero, F. , and Degasperis, A. , 1982. Spectral Transform and Solitons I . Amsterdam: North Holland; 1982.
   Google Scholar
• Dodd, R. K. , Eilbeck, J. C. , Gibbon, J. D. , and Morris, M. C. , 1982. Solitons and Nonlinear Wave Equations . London: Academic Press; 1982.
   Google Scholar
• Dorfman, I. , 1993. Dirac Structures and Integrability of Nonlinear Evolution Equations . Chichester, UK: Wiley; 1993.
   Google Scholar
• Eklund, H. , 2003. "Symbolic computation of conserved densities and fluxes for nonlinear systems of differential-difference equations". Golden, Colorado: Colorado School of Mines; 2003, M.S. thesis.
   Google Scholar
• H. Eklund and W. Hereman, DDEDensityFlux.m: A Mathematica package for the symbolic computation of conserved densities and fluxes for nonlinear systems of differential-difference equations, 2003; software available at http://inside.mines.edu/~whereman/software/DDEDensityflux.
   Google Scholar
• Fokas, A. S. , 1987. Symmetries and integrability , Stud. Appl. Math. 77 (1987), pp. 253–299.
   Web of Science ®Google Scholar
• Fokas, A. S. , and Fuchssteiner, B. , 1980. On the structure of symplectic operators and hereditary symmetries , Il Nuovo Cimento 28 (1980), pp. 229–303.
   Google Scholar
• Fuchssteiner, B. , 1979. Application of hereditary symmetries to nonlinear evolution equations , Nonlinear Anal. 3 (1979), pp. 849–862.
   Google Scholar
• Fuchssteiner, B. , and Fokas, A. S. , 1981. Symplectic structures, their Bäcklund transformations and hereditary symmetries , Phys. D 4 (1981), pp. 47–66.
   Web of Science ®Google Scholar
• Fuchssteiner, B. , and Oevel, W. , 1982. The bi-Hamiltonian structure of some nonlinear fifth- and seventh-order differential equations and recursion formulas for their symmetries and conserved covariants , J. Math. Phys. 23 (1982), pp. 358–363.
   Web of Science ®Google Scholar
• Fuchssteiner, B. , Oevel, W. , and Wiwianka, W. , 1987. Computer-algebra methods for investigation of hereditary operators of higher order soliton equations , Comput. Phys. Comm. 44 (1987), pp. 47–55.
   Web of Science ®Google Scholar
• Fuchssteiner, B. , Ivanov, S. , and Wiwianka, W. , 1997. Algorithmic determination of infinite-dimensional symmetry groups for integrable systems in 1+1 dimensions , Math. Comput. Model. 25 (1997), pp. 91–100.
   Google Scholar
• Gelfand, I. M. , and Dickey, L. A. , 1975. Asymptotic properties of the resolvent of Sturm-Liouville equations, and the algebra of Korteweg-de Vries equations , Russian. Math. Surveys. 30 (1975), pp. 77–113.
   Google Scholar
• Göktaş, Ü. , 1996. "Symbolic computation of conserved densities for systems of evolution equations". Golden, Colorado: Colorado School of Mines; 1996, M.S. thesis.
   Google Scholar
• Göktaş, Ü. , 1998. "Algorithmic computation of symmetries, invariants and recursion operators for systems of nonlinear evolution and differential-difference equations". Golden, Colorado: Colorado School of Mines; 1998, Ph.D. diss.
   Google Scholar
• Ü. Göktaş and W. Hereman, Mathematica package InvariantSymmetries.m: Symbolic computation of conserved densities and generalized symmetries of nonlinear PDEs and differential-difference equations, 1997; software available at http://library.wolfram.com/infocenter/MathSource/570 and http://inside.mines.edu/~whereman/software/ invarsym.
   Google Scholar
• Göktaş, Ü. , and Hereman, W. , 1997. Symbolic computation of conserved densities for systems of nonlinear evolution equations , J. Symbolic Comput. 24 (1997), pp. 591–621.
   Web of Science ®Google Scholar
• Göktaş, Ü. , and Hereman, W. , 1998. Computation of conservation laws for nonlinear lattices , Phys. D 132 (1998), pp. 425–436.
   Web of Science ®Google Scholar
• Göktaş, Ü. , and Hereman, W. , 1999. Algorithmic computation of higher-order symmetries for nonlinear evolution and lattice equations , Adv. Comput. Math. 11 (1999), pp. 55–80.
   Web of Science ®Google Scholar
• Göktaş, Ü. , Hereman, W. , and Erdmann, G. , 1997. Computation of conserved densities for systems of nonlinear differential-difference equations , Phys. Lett. A 236 (1997), pp. 30–38.
   Web of Science ®Google Scholar
• Hereman, W. , 2006. Symbolic computation of conservation laws of nonlinear partial differential equations in multi-dimensions , Int. J. Quantum. Chem. 106 (2006), pp. 278–299.
   Web of Science ®Google Scholar
• Hereman, W. , and Göktaş, Ü. , 1999. "Integrability tests for nonlinear evolution equations". In: Wester, M. , ed. Computer Algebra Systems: A Practical Guide . New York: Wiley; 1999. pp. 211–232.
   Google Scholar
• Hereman, W. , Göktaş, Ü. , Colagrosso, M. , and Miller, A. , 1998. Algorithmic integrability tests for nonlinear differential and lattice equations , Comput. Phys. Comm. 115 (1998), pp. 428–446.
   Web of Science ®Google Scholar
• Hereman, W. , Colagrosso, M. , Sayers, R. , Ringler, A. , Deconinck, B. , Nivala, M. , and Hickman, M. , 2005. "Continuous and discrete homotopy operators and the computation of conservation laws". In: Wang, D. , and Zheng, Z. , eds. Differential Equations with Symbolic Computation . Basel: Birkhäuser Verlag; 2005. pp. 249–285.
   Google Scholar
• Hereman, W. , Sanders, J. A. , Sayers, J. , and Wang, J. P. , 2005. "Symbolic computation of polynomial conserved densities, generalized symmetries, and recursion operators for nonlinear differential-difference equations". In: Winternitz, P. , Gomez-Ullate, D. , Iserles, A. , Levi, D. , Olver, P. J. , Quispel, R. , and Tempesta, P. , eds. Group Theory and Numerical Analysis . CRM Proc. & Lect. Ser. Vol. 39. Providence, RI: AMS; 2005. pp. 267–282.
   Google Scholar
• Hereman, W. , Adams, P. A. , Eklund, H. L. , Hickman, M. S. , and Herbst, B. M. , 2009. "Direct methods and symbolic software for conservation laws of nonlinear equations". In: Yan, Z. , ed. Advances in Nonlinear Waves and Symbolic Computation . New York: Nova Science Publishers; 2009. pp. 19–79.
   Google Scholar
• Hickman, M. , and Hereman, W. , 2003. Computation of densities and fluxes of nonlinear differential-difference equations , Proc. R. Soc. Lond. A 459 (2003), pp. 2705–2729.
   Web of Science ®Google Scholar
• R. Hirota, Direct methods in soliton theory, in Solitons, R. Bullough and P. Caudrey, eds., Topics in Current Physics, 17, ch. 5, Springer-Verlag, New York, 1980, pp. 157–176.
   Google Scholar
• Hirota, R. , 2004. "The Direct Method in Soliton Theory". Cambridge, UK: Cambridge University Press; 2004, [English translation], A. Nagai, J. Nimmo, and C. Gilson. transl. & eds.
   Google Scholar
• Hirota, R. , Grammaticos, B. , and Ramani, A. , 1986. Soliton structure of the Drinfel'd-Sokolov-Wilson equation , J. Math. Phys. 27 (1986), pp. 1499–1505.
   Web of Science ®Google Scholar
• Karasu, A. , Karasu, A. , and Sakovich, S. Yu. , 2004. A strange recursion operator for a new integrable system of coupled Korteweg-de Vries equations , Acta Appl. Math. 83 (2004), pp. 85–94, (Kalkanli).
   Web of Science ®Google Scholar
• Konopelchenko, B. G. , 1981. "Nonlinear Integrable Equations". New York, PA: Springer Verlag; 1981, Lecture Notes in Physics Vol. 270.
   Google Scholar
• Lakshmanan, M. , and Kaliappan, P. , 1983. Lie transformations, nonlinear evolution equations, and Painlevé forms , J. Math. Phys. 24 (1983), pp. 795–806.
   Web of Science ®Google Scholar
• Lamb, G. L. , 1980. "Elements of Soliton Theory". New York: Wiley; 1980.
   Google Scholar
• Lax, P. , 1968. Integrals of nonlinear equations of evolution and solitary waves , Comm. Pure Appl. Math. 21 (1968), pp. 467–490.
   Web of Science ®Google Scholar
• Magri, F. , 1978. A simple model of the integrable Hamiltonian equation , J. Math. Phys. 19 (1978), pp. 1156–1162.
   Web of Science ®Google Scholar
• Meshkov, A. G. , 2000. Computer package for investigation of the complete integrability . Presented at Proceedings of Institute of Maths of NAS of Ukraine, Part I.
   Google Scholar
• Mikhailov, A. V. , and Novikov, V. S. , 2002. Perturbative symmetry approach , J. Phys. A 35 (2002), pp. 4775–4790.
   Google Scholar
• Mikhailov, A. V. , Shabat, A. B. , and Yamilov, R. I. , 1987. The symmetry approach to the classification of non-linear equations: Complete lists of integrable systems , Uspekhi. Mat. Nauk. 42 (1987), pp. 3–53.
   Google Scholar
• Mikhailov, A. V. , Shabat, A. B. , and Sokolov, V. V. , 1991. "The symmetry approach to classification of integrable equations". In: Zakharov, V. E. , ed. What Is Integrability? . Berlin Heidelberg: Springer Verlag; 1991. pp. 115–184.
   Google Scholar
• Newell, A. C. , 1985. Solitons in Mathematics and Physics . Philadelphia, PA: SIAM; 1985, CBMS-NSF Regional Conference Series in Applied Mathematics Vol. 48.
   Google Scholar
• Olver, P. J. , 1977. Evolution equations possessing infinitely many symmetries , J. Math. Phys. 18 (1977), pp. 1212–1215.
   Web of Science ®Google Scholar
• Olver, P. J. , 1993. Applications of Lie groups to Differential Equations, . New York: Springer-Verlag; 1993, Graduate Texts in Mathematics Vol. 107.
   Google Scholar
• Praught, J. , and Smirnov, R. G. , 2005. Andrew Lenard: A mystery unraveled , Symmetry Integrability Geom. Methods Appl. (SIGMA) 1 (2005), pp. 1–7, Paper 006.
   Web of Science ®Google Scholar
• Sanders, J. , and Wang, J. P. , 1998. Combining maple and form to decide on integrability questions , Comput. Phys. Comm. 115 (1998), pp. 447–459.
   Web of Science ®Google Scholar
• Sanders, J. , and Wang, J. P. , 2001. Integrable systems and their recursion operators , Nonlinear Anal. 47 (2001), pp. 5213–5240.
   Web of Science ®Google Scholar
• Sanders, J. , and Wang, J. P. , 2001. On recursion operators , Phys. D 149 (2001), pp. 1–10.
   Web of Science ®Google Scholar
• Wang, J. P. , 1998. "Symmetries and conservation laws of evolution equations". Amsterdam: Thomas Stieltjes Institute for Mathematics; 1998, Ph.D. diss.
   Google Scholar
• Wang, J. P. , 2002. A list of (1+1) dimensional integrable equations and their properties , J. Nonlinear Math. Phys. 9 (2002), pp. 213–233.
   Web of Science ®Google Scholar
• Willox, R. , Hereman, W. , and Verheest, F. , 1995. Complete integrability of a modified vector derivative nonlinear Schrördinger equation , Phys. Scr. 52 (1995), pp. 21–26.
   Web of Science ®Google Scholar