Nonlinear physics: Integrability, chaos and beyond

Journal of the Franklin Institute

Volumn 334, Issue 5-6, 1997, Pages 909-969

  Lakshmanan, M ^a  

^a BHARATHIDASAN UNIVERSITY   (India)

Author keywords

[No Author keywords available]

Indexed keywords

CHAOS THEORY; NONLINEAR EQUATIONS; SOLITONS; SYSTEM STABILITY;

INTEGRABILITY; NONLINEAR PHYSICS;

NONLINEAR SYSTEMS;

EID: 0031220596     PISSN: 00160032     EISSN: None     Source Type: Journal    
DOI: 10.1016/s0016-0032(97)00031-8     Document Type: Article

References (94)

1. R. K. Bullough, in Solitons: Introduction and Applications, ed. M. Lakshmanan, Springer-Verlag, New York, 1988.
2. M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991.
3. S. Kovalevskaya, Acta. Math., Vol. 12, pp. 177, 1889.
4. S. Kovalevskaya, Acta. Math., Vol. 14, pp. 81, 1889.
5. M. Lakshmanan and R. Sahadevan, Physics Reports, Vol. 24, pp. 795, 1993.
6. M. D. Kruskal and P. A. Clarkson, Studies in Appl. Math., Vol. 86, pp. 87, 1992.
7. L. Brillouin, Scientific Uncertainties and Informations, Academic Press, New York, 1964.
8. P. Holmes, Physics Reports, Vol. 193, pp. 138, 1990.
9. B. Van der Pol and J. Van der Mark, Nature, Vol. 120, pp. 363, 1927.
10. E. A. Jackson, Perspectives of Nonlinear Dynamics, Vol. 1, Cambridge University Press, Cambridge, 1991.
11. M. L Cartwright and J. E. Littlewood, J. Lond. Math. Soc., Vol. 20, pp. 180, 1945.
12. N. Levinson, Ann. Math., Vol. 50, pp. 127, 1949.
13. M. Lakshmanan and K. Murali, Chaos in Nonlinear Oscillators: Controlling and Synchronization, World Scientific, Singapore, 1996.
14. H. Schiff, Proc. Roy. Soc. London, Vol. A269, pp. 277, 1962.
15. A. Einstein, The Meaning of Relativity, Princeton University Press, Princeton, 1965.
16. E. Fermi, J. Pastaand S. Ulam, Studies of Nonlinear Problem I, Los Alamos Report LA 1940, 1955.
17. N. Zabuski and M. D. Kruskal, Phys. Rev. Lett., Vol. 15, pp. 240, 1965.
18. M. Henon and C. Heils, Astrophysics. J., Vol. 69, pp. 73, 1964.
19. E. N. Lorenz, J. Atmospheric Sci., Vol. 20, pp. 130, 1963.
20. E. N. Lorenz, J. Atmospheric Sci., Vol. 20, pp. 448, 1963.
21. J. Ford, Physics Reports, Vol. 213, pp. 271, 1992.
22. C. S. Gardenar, J. M. Greene, M. D. Kruskal and R. M. Miura, Phys. Rev. Lett., Vol. 19, pp. 1095, 1967.
23. L. E. Reichl, The Transition to Chaos in Conservative Classical Systems: Quantum Manifestations, Springer-Verlag, New York, 1992.
24. A. Chados, Hadjimichael, E. and Tze, C. (ed.), Solitons in Nuclear and Elementary Particle Physics, World Scientific, Singapore, 1983.
25. T. H. R. Skyrme, Nucl. Phys., Vol. 31, pp. 556, 1962.
26. M. J. Ablowitz and H. Segur, Solitons and Inverse Scattering Transform, SIAM, Philadelphia, 1981.
27. A. S. Fokas and V. E. Zakharov (ed.), Recent Developments in Soliton Theory, Springer, Berlin, 1992.
28. M. Lakshmanan, Int. J. Bifurcation and Chaos, Vol. 3, pp. 3, 1993.
29. M. Lakshmanan (ed.), Solitons in Science and Engineering: Theory and Applications, Special issue of Chaos, Solitons and Fractals, Pergamon, New York, 1995.
30. M. Lakshmanan (ed.), Solitons:Introduction and Applications, Springer-Verlag, 1988.
31. Y. Matsuno, Bilinear Transformation Method, Academic Press, New York, 1984.
32. S. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov, Theory of Solitons, Consultants Bureau, New York, 1984.
33. C. Rogers and W. F. Shadwick, Bäcklund Transformations and Applications, Academic, New York, 1982.
34. J. Weiss, M. Tabor and G. Carnevale, J. Math. Phys., Vol. 24, pp. 522, 1983.
35. N. Asano and Y. Kato, Algebraic and Spectral Methods for Nonlinear Wave Equations, Longman Scientific and Technical, London, 1990.
36. L. A. Dickey, Soliton Equations and Hamiltonian Systems, World Scientific, Singapore, 1991.
37. G. Magnano and F. Magri, Rev. Math. Phys., Vol. 3, pp. 403, 1991.
38. M. Wadati, T. Deguchi and T. Akutsu, Physics Reports, Vol. 180, pp. 247, 1989.
39. A. Sanchez and L. Vazquez, Int. J. Mod. Phys., Vol. B5, pp. 2825, 1991.
40. A. Hasegawa, Optical Solitons in Fibers, Springer-Verlag, New York, 1989.
41. L. Lam and J. Prost, Solitons in Liquid Crystals, Springer Verlag, Berlin, 1991.
42. H. D. Wahlquist and F. B. Estabrook, J. Math. Phys., Vol. 16, pp. 1, 1975.
43. V. E. Zakharov and A. B. Shabat, Func. Anal. Appl., Vol. 8, pp. 226, 1974.
44. V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer Verlag, Berlin, 1991.
45. P. M. Santini, M. J. Ablowitz and A. S. Fokas, J. Math. Phys., Vol. 25, pp. 2614, 1984.
46. R. Hirota, Direct methods of finding exact solutions of nonlinear evolution equations. In Solitons, ed. R. K. Bullough and P. J. Caudrey, Springer-Verlag, Berlin, 1980.
47. E. Date, M. Jimbo, M. Kashiwara and T. Miwa, in Proceedings of the RIMS Symposium on Nonlinear Integrable Systems, ed. M. Jimbo and T. Miwa, World Scientific, Singapore, 1983.
48. R. M. Miura, Siam Rev., Vol. 18, pp. 412, 1976.
49. G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer-Verlag, Berlin, 1989.
50. V. E. Zakharov and L. D. Faddeev, Func. Anal. Appl., Vol. 5, pp. 280, 1971.
51. M. J. Ablowitz, A. Ramani and H. Segur, J. Math. Phys., Vol. 21, No. 715, pp. 1006, 1980.
52. M. Lakshmanan and R. Kaliappan, J. Math. Phys., Vol. 24, pp. 795, 1983.
53. M. Lakshmanan (Ed.) Special issue on Solitons in Science and Engineering: Theory and Applications Chaos, Solitons and Fractals, 5, 2213, 1995.
54. L. D. Landau and E. M. Lifshitz, Phys. Z. Sov., Vol. 8, pp. 153, 1935.
55. M. Lakshmanan, Phys. Lett., Vol. A61, pp. 53, 1977.
56. L. A. Takhtajan, Phys. Lett., Vol. 64A, pp. 235, 1977.
57. H. J. Mikeska and M. Steiner, Advances in Physics, Vol. 40, pp. 191, 1991.
58. A. M. Kosevich, B. A. Ivanov and A. S. Kovalov, Physics Reports, Vol. 195, pp. 117, 1990.
59. L. F. Mollenauer, Optics Lett., Vol. 15, pp. 1203, 1990.
60. G. P. Agarwal, Nonlinear Fibre Optics, Academic Press, New York, 1995.
61. L. Lam, Chaos, Solitons and Fractals, Vol. 5, pp. 2463, 1995.
62. R. S. Ward, Multi-dimensional integrable systems. In Field Theories, Quantum Gravity and Strings II, ed. H. J. de Vega and N. Sanchez, Lecture Notes in Physics 280, Springer-Verlag, Berlin, 1986.
63. R. Ward, Phil. Trans. R. Soc. Lond., Vol. A315, pp. 451, 1985.
64. E. Witten, Nucl. Phys., Vol. B249, pp. 557, 1985.
65. V. E. Zakharov and S. V. Manakov, Func. Anal. Appl., Vol. 19, pp. 89, 1985.
66. A. S. Fokas and P. M. Santini, Physica, Vol. D44, pp. 99, 1990.
67. R. Beals and P. R. Coifman, Inverse Problems, Vol. 5, pp. 87, 1989.
68. A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion, Springer-Verlag, New York, 1983.
69. P. G. Drazin, Nonlinear Systems, Cambridge University Press, Cambridge, 1992.
70. J. L. McCauley, Chaos, Dynamics and Fractals, Cambridge University Press, Cambridge, 1993.
71. T. Mullin (ed.), The Nature of Chaos, Clasendar Press, Oxford, 1993.
72. J. Gleick, Chaos: Making of a New Science, Cardinal, 1987.
73. M. C. Gutzwiller, Chaos and Quantum Mechanics, Springer-Verlag, Berlin, 1990.
74. K. Nakamura, Quantum Chaos: A new paradigm of Nonlinear Dynamics, Cambridge University Press, Cambridge, 1993.
75. T. Shinbrot, G. Grebogi and J. A. Yorke, Nature, Vol. 363, pp. 411, 1993.
76. L. M. Pecora and T. L. Carrol, Phys. Rev. Lett., Vol. 64, pp. 821, 1990.
77. P. L. Sachdev, Nonlinear Diffusive Waves, Cambridge University Press, Cambridge, 1987.
78. A. S. Fokas and Yorstos, SIAM J. Appl. Math., Vol. 42, pp. 318, 1982.
79. A. R. Ramani, B. G. Grammaticos and T. C. Bountis, Phys. Reports, Vol. 180, pp. 169, 1989.
80. T. C. Bountis, Int. J. Bifurcation and Chaos, Vol. 2, pp. 217, 1992.
81. R. Rajaraman, Solitons and Instantons, North-Holland, Amsterdam, 1982.
82. V. G. Makhankov, V. P. Rybakov and V. I. Sanyuk, The Skyrme Model, Springer, Berlin, 1993.
83. R. Radha and M. Lakshmanan, J. Phys., Vol. A 29, pp. 1551, 1996.
84. R. Scharf, Chaos, Solitons and Fractals, Vol. 5, pp. 2527, 1995.
85. Y. S. Kivshar and K. H. Spatchek, Chaos, Solitons and Fractals, Vol. 5, pp. 2551, 1990.
86. W. F., Jr. Brown, Micromagnetics, Wiley, New York, 1963.
87. M. Lakshmanan and K. Nakamur, Phys. Rev. Lett., Vol. 53, pp. 2497, 1984.
88. R. K. Bullough and P. J. Caudrey (ed.), Solitons, Springer-Verlag, Berlin, 1980.
89. P. A. Clarkson, M. J. Ablowitz and A. S. Fokas, The inverse scattering transform for multidimensional (2+1) problems. In Lecture notes in Physics 189, Springer Verlag, Berlin, 1983.
90. A. S. Fokas and M. J. Ablowitz, Stud. Appl. Math., Vol. 69, pp. 211, 1983.
91. M. Lakshmanan and R. K. Bullough, Phys. Lett., Vol. 80A, pp. 287, 1980.
92. B. Piette and R. Zakrzewski, Chaos, Solitons and Fractals, Vol. 5, pp. 2495, 1995.
93. R. Scharf, Chaos, Solitons and Fractals, Vol. 5, pp. 2527, 1995.
94. V. E. Zakharov and L. A. Takhtajan, Theor. Math. Phys., Vol. 38, pp. 117, 1979.