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A symbolic algorithm for computing recursion operators of nonlinear partial differential equations. (English) Zbl 1191.65166

Summary: A recursion operator is an integro-differential operator which maps a generalized symmetry of a nonlinear partial differential equation (PDE) to a new symmetry. Therefore, the existence of a recursion operator guarantees that the PDE has infinitely many higher-order symmetries, which is a key feature of complete integrability. Completely integrable nonlinear PDEs have a bi-Hamiltonian structure and a Lax pair; they can also be solved with the inverse scattering transform and admit soliton solutions of any order.
A straightforward method for the symbolic computation of polynomial recursion operators of nonlinear PDEs in \((1+1)\) dimensions is presented. Based on conserved densities and generalized symmetries, a candidate recursion operator is built from a linear combination of scaling invariant terms with undetermined coefficients. The candidate recursion operator is substituted into its defining equation and the resulting linear system for the undetermined coefficients is solved.
The method is algorithmic and is implemented in Mathematica. The resulting symbolic package PDERecursionOperator.m can be used to test the complete integrability of polynomial PDEs that can be written as nonlinear evolution equations. With PDERecursionOperator.m, recursion operators were obtained for several well-known nonlinear PDEs from mathematical physics and soliton theory.

MSC:

65P10 Numerical methods for Hamiltonian systems including symplectic integrators
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
68W30 Symbolic computation and algebraic computation
35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
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References:

[1] DOI: 10.1017/CBO9780511623998 · doi:10.1017/CBO9780511623998
[2] Ablowitz M. J., Solitons and the Inverse Scattering Transform (1981) · Zbl 0472.35002
[3] DOI: 10.1063/1.524491 · Zbl 0445.35056 · doi:10.1063/1.524491
[4] 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. M.S. thesis
[5] P.J. Adams and W. Hereman,TransPDEDensityFlux.m: AMathematicapackage for the symbolic computation of conserved densities and fluxes for systems of partial differential equations with transcendental nonlinearities, 2002; software available athttp://inside.mines.edu/ whereman/software/TransPDEDensityFlux
[6] 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. M.S. thesis
[7] D. Baldwin and W. Hereman,PainleveTest.m: AMathematicapackage for the Painlevé test of systems of ODEs and PDEs, 2003; software available athttp://inside.mines.edu/ whereman/software/painleve/mathematica
[8] DOI: 10.2991/jnmp.2006.13.1.8 · Zbl 1110.35300 · doi:10.2991/jnmp.2006.13.1.8
[9] D. Baldwin and W. Hereman,PDERecursionOperator.m:A Mathematicapackage for the symbolic computation of recursion operators for nonlinear partial differential equations, 2003–2009; software available athttp://inside.mines.edu/ whereman/software/PDERecursionOperator
[10] Baldwin D., CRM Proc. & Lect. Ser. Vol. 39, in: Group Theory and Numerical Analysis pp 17– (2005)
[11] DOI: 10.1088/0305-4470/26/24/024 · Zbl 0835.35134 · doi:10.1088/0305-4470/26/24/024
[12] Calogero F., Spectral Transform and Solitons I (1982)
[13] Dodd R. K., Solitons and Nonlinear Wave Equations (1982) · Zbl 0496.35001
[14] Dorfman I., Dirac Structures and Integrability of Nonlinear Evolution Equations (1993) · Zbl 0717.58026
[15] Eklund, H. 2003. ”Symbolic computation of conserved densities and fluxes for nonlinear systems of differential-difference equations”. Golden, Colorado: Colorado School of Mines. M.S. thesis
[16] H. Eklund and W. Hereman,DDEDensityFlux.m: AMathematicapackage for the symbolic computation of conserved densities and fluxes for nonlinear systems of differential-difference equations, 2003; software available athttp://inside.mines.edu/ whereman/software/DDEDensityflux
[17] Fokas A. S., Stud. Appl. Math. 77 pp 253– (1987) · Zbl 0639.35075 · doi:10.1002/sapm1987773253
[18] Fokas A. S., Il Nuovo Cimento 28 pp 229– (1980)
[19] DOI: 10.1016/0362-546X(79)90052-X · Zbl 0419.35049 · doi:10.1016/0362-546X(79)90052-X
[20] DOI: 10.1016/0167-2789(81)90004-X · Zbl 1194.37114 · doi:10.1016/0167-2789(81)90004-X
[21] DOI: 10.1063/1.525376 · Zbl 0489.35029 · doi:10.1063/1.525376
[22] DOI: 10.1016/0010-4655(87)90015-4 · Zbl 0664.65118 · doi:10.1016/0010-4655(87)90015-4
[23] DOI: 10.1016/S0895-7177(97)00061-7 · Zbl 0882.58025 · doi:10.1016/S0895-7177(97)00061-7
[24] DOI: 10.1070/RM1975v030n05ABEH001522 · Zbl 0334.58007 · doi:10.1070/RM1975v030n05ABEH001522
[25] Göktaş, Ü. 1996. ”Symbolic computation of conserved densities for systems of evolution equations”. Golden, Colorado: Colorado School of Mines. M.S. thesis
[26] 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. Ph.D. diss
[27] Ü. Göktaş and W. Hereman,MathematicapackageInvariantSymmetries.m: Symbolic computation of conserved densities and generalized symmetries of nonlinear PDEs and differential-difference equations, 1997; software available athttp://library.wolfram.com/infocenter/MathSource/570andhttp://inside.mines.edu/ whereman/software/invarsym
[28] DOI: 10.1006/jsco.1997.0154 · Zbl 0891.65129 · doi:10.1006/jsco.1997.0154
[29] DOI: 10.1016/S0167-2789(98)00140-7 · Zbl 0940.34065 · doi:10.1016/S0167-2789(98)00140-7
[30] DOI: 10.1023/A:1018955405327 · Zbl 0936.65123 · doi:10.1023/A:1018955405327
[31] DOI: 10.1016/S0375-9601(97)00750-0 · Zbl 0969.35542 · doi:10.1016/S0375-9601(97)00750-0
[32] DOI: 10.1002/qua.20727 · Zbl 1188.68364 · doi:10.1002/qua.20727
[33] Hereman W., Computer Algebra Systems: A Practical Guide pp 211– (1999)
[34] DOI: 10.1016/S0010-4655(98)00121-0 · doi:10.1016/S0010-4655(98)00121-0
[35] Hereman W., Differential Equations with Symbolic Computation pp 249– (2005)
[36] Hereman W., CRM Proc. & Lect. Ser. Vol. 39, in: Group Theory and Numerical Analysis pp 267– (2005)
[37] Hereman W., Advances in Nonlinear Waves and Symbolic Computation pp 19– (2009)
[38] DOI: 10.1098/rspa.2003.1151 · Zbl 1059.35154 · doi:10.1098/rspa.2003.1151
[39] R. Hirota,Direct methods in soliton theory, inSolitons, R. Bullough and P. Caudrey, eds., Topics in Current Physics, 17, ch. 5, Springer-Verlag, New York, 1980, pp. 157–176
[40] DOI: 10.1017/CBO9780511543043 · doi:10.1017/CBO9780511543043
[41] DOI: 10.1063/1.527110 · Zbl 0638.35071 · doi:10.1063/1.527110
[42] DOI: 10.1023/B:ACAP.0000035590.28726.b3 · Zbl 1062.37073 · doi:10.1023/B:ACAP.0000035590.28726.b3
[43] Konopelchenko, B. G. 1981. ”Nonlinear Integrable Equations”. New York, PA: Springer Verlag. Lecture Notes in Physics Vol. 270 · Zbl 0469.35009
[44] DOI: 10.1063/1.525752 · Zbl 0525.35022 · doi:10.1063/1.525752
[45] Lamb, G. L. 1980. ”Elements of Soliton Theory”. New York: Wiley. · Zbl 0445.35001
[46] DOI: 10.1002/cpa.3160210503 · Zbl 0162.41103 · doi:10.1002/cpa.3160210503
[47] DOI: 10.1063/1.523777 · Zbl 0383.35065 · doi:10.1063/1.523777
[48] Meshkov, A. G. Computer package for investigation of the complete integrability. Proceedings of Institute of Maths of NAS of Ukraine. Edited by: Nikitin, A. G. and Boyko, V. M. Vol. 30, pp.35–46. Kyiv: Institute of Mathematics. Part I · Zbl 0951.68194
[49] DOI: 10.1088/0305-4470/35/22/309 · Zbl 1039.35008 · doi:10.1088/0305-4470/35/22/309
[50] Mikhailov A. V., Uspekhi. Mat. Nauk. 42 pp 3– (1987)
[51] Mikhailov A. V., What Is Integrability? pp 115– (1991) · doi:10.1007/978-3-642-88703-1_4
[52] Newell A. C., Solitons in Mathematics and Physics (1985) · Zbl 0565.35003 · doi:10.1137/1.9781611970227
[53] DOI: 10.1063/1.523393 · Zbl 0348.35024 · doi:10.1063/1.523393
[54] Olver P. J., Applications of Lie groups to Differential Equations,, 2. ed. (1993) · Zbl 0785.58003 · doi:10.1007/978-1-4612-4350-2
[55] Praught J., Symmetry Integrability Geom. Methods Appl. (SIGMA) 1 pp 1– (2005)
[56] DOI: 10.1016/S0010-4655(98)00122-2 · Zbl 1001.65132 · doi:10.1016/S0010-4655(98)00122-2
[57] DOI: 10.1016/S0362-546X(01)00630-7 · Zbl 1042.37531 · doi:10.1016/S0362-546X(01)00630-7
[58] DOI: 10.1016/S0167-2789(00)00188-3 · Zbl 0972.35137 · doi:10.1016/S0167-2789(00)00188-3
[59] Wang, J. P. 1998. ”Symmetries and conservation laws of evolution equations”. Amsterdam: Thomas Stieltjes Institute for Mathematics. Ph.D. diss
[60] DOI: 10.2991/jnmp.2002.9.s1.18 · Zbl 1362.37137 · doi:10.2991/jnmp.2002.9.s1.18
[61] DOI: 10.1088/0031-8949/52/1/003 · doi:10.1088/0031-8949/52/1/003
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