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Bäcklund transformations of the first kind associated with Monge-Ampère’s equations

Published online by Cambridge University Press:  22 January 2016

Michihiko Matsuda
Michihiko Matsuda*
Affiliation:
Department of Mathematics, Osaka University Toyonaka, Osaka, 560, Japan
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Due to Clairin and Goursat, a Bäcklund transformation of the first kind can be associated with Monge-Ampère’s equation. We shall consider Monge-Ampère’s equation of the form s + f(x, y, z, p, q) + g(x, y, z, p, q) t = 0, where p = ∂z/∂x, q = ∂z/∂y, s = ∂2z/∂x∂y, t = ∂2z/∂y2. The following theorems will be obtained:

1. The transformed equation takes on the same form s′ + f′ + g′t′ = 0 if and only if the given equation can be transformed to a Teixeira equation s + L(x, y, z, q)t + M(x, y, z, q)p + N(x, y, z, q) = 0 by a contact transformation.

2. Teixeira equation s + tL + pM + N = 0 is solved by integrable systems of order n if and only if the transformed equation is solved by integrable systems of order n — 1.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 51 , September 1973 , pp. 161 - 184
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1973

References

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