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On some geometry associated with a generalised Toda lattice

Published online by Cambridge University Press:  17 April 2009

P.J. Vassiliou
P.J. Vassiliou
Affiliation:
Faculty of Information Sciences and EngineeringUniversity of CanberraBelconnen ACT 2616, Australia
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We define the notion of Darboux integrability for linear second order partial differential operators,

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We then build on certain geometric results of E. Vessiot related to the theory of Monge characteristics to show that the Darboux integrable operators L can be used to obtain a solution of the A2 Toda field theory. This solution is parametrised by four arbitrary functions. The approach presented in this paper thereby represents an alternative means of linearising the A2 Toda equations and may be contrasted with the known linearisation via the Lax pair.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 49 , Issue 3 , June 1994 , pp. 439 - 462
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Bryant, R., Chern, S.S., Gardner, R.B., Goldschmidt, H. and Grifiths, P., Exterior differential systems, MSIR series (Springer-Verlag, Berlin, Heidelberg, New York, 1989).Google Scholar
[2]Darboux, G., Lecons sur la theorie generale de surface 2 (Gauthier-Villars, Paris, 1889).Google Scholar
[3]Gardner, R.B., The method of equivalence and its applications, SIAM CBMS — NSF Regional Conference Series, 1989.Google Scholar
[4]Gardner, R.B. and Karman, N., ‘Characteristics and the geometry of hyperbolic equations in the plane’, J. Differential Equations (to appear).Google Scholar
[5]Grundland, A.M. and Vassiliou, P.J., ‘On the solvability of the Cauchy problem for Riemann double waves by the method of Monge - Darboux’, Analysis 11 (1992), 221–278.Google Scholar
[6]Grundland, A.M. and Vassiliou, P.J., ‘Riemann double waves, Darboux method and the Painleve property’, in Proceedings of the NATO workshop on Painleve transcendents, their asymptotics and physical applications, (Levi, D. and Winternitz, P., Editors) (Plenum Press, 1992).Google Scholar
[7]Hermann, R., The geometry of nonlinear differential equations, Interdisciplinary Mathematics 12, Backlund Transformations and Solitons Part A (Math Science Press, Brookline, MA., 1976).Google Scholar
[8]Leznov, A.N. and Saveliev, M.V., ‘Representation of zero curvature for the system of nonlinear partial differential equations xα, zx¯ = exp(kx)α and its integrability’, Lett. Math. Phys. 3 (1979), 489–494.CrossRefGoogle Scholar
[9]Stormark, O., ‘Formal and local solvability of partial differential equations’, Royal Swedish Institute of Technology (preprint).Google Scholar
[10]Vassiliou, P.J., ‘On local equivalence for vector field systems’, Bull. Austral. Math. Soc. 42 (1990), 215–229.CrossRefGoogle Scholar
[11]Vassiliou, P.J., in preparation.Google Scholar
[12]Vessiot, E., ‘Sur les equations aux derives partielles du second ordre F(x, y, z, p, q, r, s, t) = 0 integrable par la methode de Darboux’, J. Math. Pures Appl. 18 (1939), 1–61.J. Math. Pures Appl., 21 1–66, (1942).Google Scholar
[13]Weiss, J., Backland transformations and the Painleve property, (preprint 509, 04 1989) (Institute for Mathematics and its Applications, University of Minnesota).Google Scholar