Algorithmic integrability tests for nonlinear differential and lattice equations

Computer Physics Communications

Volumn 115, Issue 2-3, 1998, Pages 428-446

  Hereman, Willy ^a   Goktaş, Ünal ^a   Colagrosso, Michael D ^a   Miller, Antonio J ^b  

^a Department of Metallurgical and Materials Engineering   (United States)

^b PENNSYLVANIA STATE UNIVERSITY   (United States)

Author keywords

Conservation law; Differential equation; Integrability; Invariant; Lattice; Painlev test; Symmetry

Indexed keywords

ALGORITHMS; COMPUTER AIDED ANALYSIS; DIFFERENCE EQUATIONS; POLYNOMIALS; TESTING;

INTEGRABILITY TESTING;

DIFFERENTIAL EQUATIONS;

EID: 0032477111     PISSN: 00104655     EISSN: None     Source Type: Journal    
DOI: 10.1016/s0010-4655(98)00121-0     Document Type: Article

References (66)

1. M. Lakshmanan, Nonlinear Physics: Integrability, Chaos and Beyond, Preprint, Center for Nonlinear Dynamics & Department of Physics, Bharathidasan University, Tiruchirapalli 620 024, India (1997).
2. R.S. Palais, Bull. Am. Math. Soc. 34 (1997) 339.
3. S. Rauch-Wojciechowski, in: Soliton Theory. A Survey of Results, A. Fordy, ed. (Manchester University Press, Manchester, 1990) p. 235.
4. M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Math. Soc. Lec. Note Ser. 149 (Cambridge Univ. Press, London, 1991).
5. A.S. Fokas, Stud. Appl. Math. 77 (1987) 253.
6. A.V. Mikhailov, A.B. Shabat, V.V. Sokolov, in: What Is Integrability?, V.E. Zakharov, ed. (Springer, Berlin, 1991) p. 115.
7. H. Flaschka, A.C. Newell, M. Tabor, in: What is Integrability, V.E. Zakharov, ed. (Springer, New York, 1991) p. 73.
8. S.P. Novikov, The Mathematical Intelligencer 14 (1992) 13.
9. B. Grammaticos, A. Ramani, in: Integrability of Nonlinear Systems, Y. Kosmann-Schwarzbach, B. Grammaticos, K.M. Tamizhmani, eds. (Springer, Berlin, 1997) p. 30.
10. D. Levi, P. Winternitz, eds., Painlevé Transcendents: Their Asymptotics and Physical Applications, NATO Adv. Sci. Inst. Ser. B (Phys.), Vol. 278 (Plenum Press, New York, 1992).
11. F. Cariello, M. Tabor, Physica D 39 (1989) 77.
12. R. Conte, in: Introduction to Methods of Complex Analysis and Geometry for Classical Mechanics and Non-Linear Waves, D. Benest, C. Frœschlé, eds. (Éditions Frontières, Gif-sur-Yvette, France, 1993) p. 49.
13. M.D. Kruskal, P.A. Clarkson, Stud. Appl. Math. 86 (1992) 87.
14. M.D. Kruskal, N. Joshi, R. Halburd, in: Integrability of Nonlinear Systems Y. Kosmann-Schwarzbach, B. Grammaticos, K.M. Tamizhmani, eds. (Springer, Berlin, 1997) p. 171.
15. M. Lakshmanan, R. Sahadevan, Phys. Rep. 224 (1993) 1.
16. M. Musette, R. Conte, J. Math. Phys. 32 (1991) 1450.
17. A.C. Newell, M. Tabor, Y.B. Zeng, Physica D 29 (1987) 1.
18. P.A. Clarkson, ed., Applications of Analytic and Geometrical Methods to Nonlinear Differential Equations, NATO Adv. Sci. Inst. Ser. C, Vol. 413 (Kluwer Academic Publishers, Dordrecht, 1993).
19. R. Conte, ed., The Painlevé Property, One Century Later, CRM Series in Mathematical Physics (Springer, Berlin, 1998).
20. W.-H. Steeb, N. Euler, Nonlinear Evolution Equations and Painlevé Test (World Scientific, Singapore, 1988).
21. J.D. Gibbon, P. Radmore, M. Tabor, D. Wood, Stud. Appl. Math. 72 (1985) 39.
22. M. Musette, R. Conte, J. Phys. A 27 (1994) 3895.
23. G. Contopoulos, B. Grammaticos, A. Ramani, J. Phys. A 28 (1995) 5313.
24. A. Ramani, B. Grammaticos, T. Bountis, Phys. Rep. 180 (1989) 159.
25. R. Conte, A.P. Fordy, A. Pickering, Physica D 69 (1993) 33.
26. C. Scheen, Theor. Comput. Sci. 187 (1997) 87.
27. W. The Macsyma programs painsing.max, painsys.max, and the Mathematica programs painsing.m, painsys.m are available via anonymous FTP from mines.edu in directory pub/papers/math_cs_dept/software; or via Internet URL: http://www.mines.edu/fs_home/whereman/.
28. Macsyma Mathematics and System Reference Manual, 15th Edition (Macsyma Inc., Arlington, MA, 1995).
29. S. Wolfram, The Mathematica book, 3rd ed. (Wolfram Media, Urbana-Champaign, Illinois & Cambridge University Press, London, 1996).
30. M.D. Kruskal, in: Painlevé Transcendents, D. Levi, P. Winternitz, eds. (Plenum Press, New York, 1992) p. 187.
31. W. Hereman, S. Angenent, MACSYMA Newsletter 6 (1989) 11.
32. W. Hereman, W. Zhuang, Acta Appl. Math. 39 (1995) 361.
33. A. Ramani, B. Dorizzi, B. Grammaticos, Phys. Rev. Lett. 49 (1982) 1539.
34. A. Goriely, J. Math. Phys. 33 (1992) 2728.
35. B. Grammaticos, A. Ramani, V. Papageorgiou, Phys. Rev. Lett. 67 (1991) 1825.
36. B. Grammaticos, V. Papageorgiou, A. Ramani, Acta Appl. Math. 39 (1995) 335.
37. I. Cherdantsev, R. Yamilov, in: Symmetries and Integrability of Difference Equations, D. Levi, L. Vinet, P. Winternitz, eds. CRM Proceedings and Lecture Notes, Vol. 9 (American Mathematical Society, Providence, RI, 1996) p. 53.
38. D. Levi, R. Yamilov, Conditions for the existence of higher symmetries of evolutionary equations on the lattice, J. Math. Phys. 38 (1997) 6648.
39. A.B. Shabat, R.I. Yamilov, Leningrad Math. J. 2 (1991) 377.
40. Ü. Göktaş, W. Hereman, J. Symb. Comput. 24 (1997) 591.
41. Ü. Göktaş, W. Hereman, Computation of conservation laws for nonlinear lattices, Physica D 123 (1998) 425.
42. Ü. Göktaş, W. Hereman, Computation of higher-order symmetries for nonlinear evolution and lattice equations, Adv. Comput. Math. (1998), submitted.
43. Ü. Göktaş, W. Hereman, Package InvariantsSymmetries.m is available at http://www.mathsource.com/cgi-bin/msitem?0208-932; MathSource is the electronic library of Wolfram Research, Inc. (Champaign, IL).
44. M.J. Ablowitz, A. Ramani, H. Segur, J. Math. Phys. 21 (1980) 715.
45. M.J. Ablowitz, A. Ramani, H. Segur, J. Math. Phys. 21 (1980) 1006.
46. J. Weiss, M. Tabor, G. Carnevale, J. Math. Phys. 24 (1983) 522.
47. M.D. Kruskal, A. Ramani, B. Grammaticos, in: Partially Integrable Evolution Equations in Physics, R. Conte, N. Boccara, eds. (Kluwer Academic Publishers, Dordrecht, 1990) p. 321.
48. D.V. Chudnovsky, G.V. Chudnovsky, M. Tabor, Phys. Lett. A 97 (1983) 268.
49. E.V. Doktorov, S.Yu. Sakovich, J. Phys. A 18 (1985) 3327.
50. R. Hirota, B. Grammaticos, A. Ramani, J. Math. Phys. 27 (1986) 1499.
51. K.M. Tamizhmani, R. Sahadevan, J. Phys. A 18 (1985) L1067.
52. F. Calogero, A. Degasperis, Spectral Transform and Solitons I (North-Holland, Amsterdam, 1982).
53. P.A. Clarkson, IMA J. Appl. Math. 44 (1990) 27.
54. W. Hereman, in: Advances in Computer Methods for Partial Differential Equations VII, R. Vichnevetsky, D. Knight, G. Richter, eds. (IMACS, New Brunswick, New Jersey, 1993) p. 326.
55. C. Sparrow, The Lorenz Equation: Bifurcations, Chaos, and Strange Attractors, Applied Mathematical Sciences 41 (Springer, New York, 1982).
56. R.L. Sachs, Physica D 30 (1988) 1.
57. C.R. Menyuk, H.H. Chen, Y.C. Lee, Phys. Rev. A 27 (1983) 1597.
58. F. Verheest, W. Hereman, H. Serras, Mon. Not. Roy. Astr. Soc. 245 (1990) 392.
59. R.M. Miura, C.S. Gardner, M.D. Kruskal, J. Math. Phys. 9 (1968) 1204.
60. P.J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed. (Springer, New York, 1993).
61. W. Hereman, Ü. Göktaş, Symbolic integrability tests for systems of nonlinear evolution equations, chapter for: Computer Algebra Systems: A Practical Guide, M. Wester, ed. (Wiley, New York, 1999), to appear.
62. Ü. Göktaş, W. Hereman, G. Erdmann, Phys. Lett. A 236 (1997) 30.
63. F. Verheest, in: Proc. Fourth Int. Conf. on Mathematical and Numerical Aspects of Wave Propagation, J.A. DeSanto, ed. (SIAM, Philadelphia, 1998) p. 398.
64. L.D. Faddeev, L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer, Berlin, 1987).
65. Y. Itoh, Prog. Theor. Phys. 78 (1987) 507.
66. J.M. Nunes da Costa, C.-M. Marle, J. Phys. A 30 (1997) 7551.