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The general symmetry algebra structure of the underdetermined equation \(u_ x=(v_{xx})^ 2\). (English) Zbl 0738.34007

Summary: In a recent paper, I. M. Anderson, N. Kamran and P. Olver [Interior, exterior, and generalized symmetries, preprint (1990)] obtained the first- and second-order generalized symmetry algebra for the system \(u_ x=(v_{xx})^ 2\), leading to the noncompact real form of the exceptional Lie algebra \(G_ 2\). Here, the structure of the general higher-order symmetry algebra is obtained. Moreover, the Lie algebra \(G_ 2\) is obtained as ordinary symmetry algebra of the associated first-order system. The general symmetry algebra for \(u_ x=f(u,v,v_ x,\dots,)\) is also established.

MSC:

34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
58J70 Invariance and symmetry properties for PDEs on manifolds
35A30 Geometric theory, characteristics, transformations in context of PDEs
17B66 Lie algebras of vector fields and related (super) algebras
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References:

[1] I. M. Anderson, N. Kamran, and P. Olver, ”Interior, exterior and generalized symmetries,” preprint (1990).
[2] P. H. M. Kersten,Symmetries: A Computational Approach(Center for Mathematics and Computer Science, Amsterdam, The Netherlands), CWI tract 34. · Zbl 0648.68052
[3] F. A. E. Pirani, D. C. Robinson, and W. F. Shadwiek,Local Jet Bundle Formulation of Bäcklund Transformations, Mathematical Physics Studies 1 (Reidel, Boston, 1979).
[4] P. J. Olver,Applications of Lie-groups to Differential Equations, Graduate Texts in Mathematics, 107 (Springer-Verlag, New York, 1986). · Zbl 0588.22001
[5] I. S. Krasilshchik, V. V. Lychagin, and A. M. Vinogradov,Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Advanced Studies in Contemporary Mathematics, Vol. 1 (Gordon and Breach, New York, 1985).
[6] R. L. Anderson and N. H. Ibragimov,Lie-Bäcklund Transformations in Applications, SI AM Studies in Applied Mathematics 1 (SI AM, Philadelphia, 1978). · Zbl 0424.53004
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