Abstract
Some evolution equations possess infinite-dimensional prolongation Lie algebras which can be made finite-dimensional by using a bigger (non-Archimedean) field. The advantage of this is that convergence problems hardly exist in such a field. Besides that, the accompanying Lie groups can be easily constructed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Wahlquist, H. D. and Estabrook, F. B., ‘Prolongation Structures of Nonlinear Evolution Equations I’, J. Math. Phys. 16, 1–7 (1975).
Bourbaki, N., Groupes et algèbres de Lie, Chap. II: ‘Algèbres de Lie libres’, Hermann, Paris, 1972.
Ibid., Chap. III: ‘Groupes de Lie’.
Van Eck, H.N., ‘The Explicit Form of the Lie Algebra of Wahlquist and Estabrook. A Presentation Problem’, Proc. Kon. Ned. Akad. Wetensch., Series A 86, 149–164 (1983).
Van Eck, H. N., Gragert, P. K. H., and Martini, R., ‘The Explicit Structure of the Nonlinear Schrödinger Prolongation Algebra’, Proc. Kon. Ned. Akad. Wetensch., Series A 86, 165–172 (1983).
Miura, R. M., Gardner, C. S., and Kruskal, M. D., ‘Korteweg-de Vries Equations and Generalizations, II. Existence of Conservation Laws and Constants of Motion’, J. Math. Phys. 9, 1204–1209 (1968).
Miura, R. M., ‘Korteweg-de Vries Equation and Generalizations, I. A. Remarkable Explicit Nonlinear Transformation’, J. Math. Phys. 9, 1202–1204 (1968).
Lax, P. D., ‘Nonlinear Partial Differential Equations of Evolution’, Actes, Congrès Intern. Math.: 1970, Vol. 2, pp. 831–840.
Lax, P. D., ‘Periodic Solutions of the KdV Equations’, in A. C. Newell (ed.), Nonlinear Wave Motion, AMS Proc., 1974, Lect. in Appl. Math., Vol. 15, pp. 85–96.
Lax, P. D., ‘Almost Periodic Solutions of the KdV Equation’, SIAM Rev. 18, 351–375 (1976).
Bourbaki, N., Variétés différentielles et analytiques, Hermann, Paris, 1971.
Estabrook, F. B. and Wahlquist, H. D., ‘Prolongation Structures of Nonlinear Evolution Equations II’, J. Math. Phys. 17, 1293–1297 (1976).
Kumei, S., ‘Group Theoretic Aspects of Conservation Laws of Nonlinear Dispersive Waves: KdV Type Equations and Nonlinear Schrödinger Equations’, J. Math. Phys. 18, 256–264 (1977).
Kaup, D. J., ‘The Estabrook-Wahlquist Method with Examples of Application’, Physica D 1D, 391–411 (1980).
Krasilshchik, I. S. and Vinogradov, A. M.: ‘Nonlocal symmetries and the Theory of Coverings: An Addendum to A. M. Vinogradov's “Local Symmetries and Conservation Laws”’, Acta Appl. Math. 2, 79–96 (1984).
Gelfand, I. M. and Dikii, L. A., ‘Asymptotic Behaviour of the Resolvent of Sturm-Liouville Equations and the Algebra of the Korteweg-de Vries Equations’, London Mathematical Society, Lecture Note Series 60, Cambridge University Press, 1981, pp. 13–49.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Van Eck, H.N. A non-Archimedean approach to prolongation theory. Lett Math Phys 12, 231–239 (1986). https://doi.org/10.1007/BF00416513
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00416513