Abstract
I began this lecture by posing the question of what the best way is to define integrability. The most promising definition seems to be one based on an associated overdetermined linear system which is “allowable”. The linear systems described in sections 3 and 4 are certainly allowable, and most known integrable equations can be obtained in this way. But there are exceptions, such as the KP equation, so the question is not yet settled. With this sort of definition, it would be difficult to establish whether or not a given equation was integrable. But one could try to classify all the integrable equations which arose from a certain type of linear system.
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References
Adler, M. & P. van Moerbeke 1980 Advances in Math. 38, 267–379.
Athorne, C. & A.P.Fordy 1986 Generalized KdV equations associated with symmetric spaces. (Leeds preprint).
Chudnovsky, D. & G. 1984 Note on Eisenstein's system of differential equations.In: Classical and Quantum Models and Arithmetic Problems, eds. D. & G. Chudnovsky (Marcel Dekker), pp. 99–115.
Date, E., M. Kashiwara, M. Jimbo & T. Miwa 1983 Transformation groups for soliton equations.In: Proceedings of RIMS Symposium on Non-Linear Integrable Systems, eds. M. Jimbo & T. Miwa (World Scientific), pp. 39–119.
Dubois-Violette, M. 1982 Phys. Lett. B 119, 157–161.
Eckhardt, B., J. Ford & F. Vivaldi 1984 Physica D 13, 339–356.
Fordy, A.P. & P.P. Kulish 1983 Comm. Math. Phys. 89, 427–443.
Forgacs, P. & N.S. Manton 1980 Comm. Math. Phys. 72, 15–35.
Goldschmidt,Y.Y. & E. Witten 1980 Phys. Lett. B 91, 392–396.
Olive, D. & N. Turok 1983 Nucl. Phys. B 215, 470–494.
Penrose, R. 1976 Gen. Rel. Grav. 7, 31–52.
Savvidy, G.K. 1984 Nucl. Phys. B 246, 302–334.
van Moerbeke, P. 1985 Phil. Trans. R. Soc. Lond. A 315, 379–390.
Ward, R.S. 1983 Gen. Rel. Grav. 15, 105–109.
Ward, R.S. 1984 Phys. Lett. A 102, 279–282.
Ward, R.S. 1984b Nucl. Phys. B 236, 381–396.
Ward, R.S. 1985 Phys. Lett. A 112, 3–5.
Wilson, G. 1985 Phil. Trans. R. Soc. Lond. A 315, 393–404.
Witten, L. 1979 Phys. Rev. D 19, 718–720.
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Ward, R.S. (1987). Multi-dimensional integrable systems. In: de Vega, H.J., Sánchez, N. (eds) Field Theory, Quantum Gravity and Strings II. Lecture Notes in Physics, vol 280. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-17925-9_33
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DOI: https://doi.org/10.1007/3-540-17925-9_33
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