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

## Extract

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.

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## References

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[1] Cartan, E., Sur l’intégration des systèmes d’équations aux différentielles totales, Ann. Sci. Ecole Norm. Sup., 18 (1901), 241311.
[2] Cartan, E., Les systèmes différentiels extérieurs et leurs applications géométriques, Hermann, Paris, 1946.
[3] Clairin, J., Sur les transformations de Bäcklund, Ann. Sci. Ecole Norm. Sup., 19 (Suppl.) (1902), 363.
[4] Forsyth, A. R., Theory of differential equations, Part I, Exact equations and Pfaff’s problem, Cambridge Univ. Press, London, 1890.
[5] Goursat, E., Le problème de Bäcklund, Memor. Sci. Math., fasc. VI, Gauthier-Villars, Paris, 1925.
[6] Lie, S., Theorie der Transformationsgruppen, II, Teubner, Leipzig, 1890.
[7] Matsuda, M., Two methods of integrating Monge-Ampère’s equations, Trans. Amer. Math. Soc., 150 (1970), 327343.
[8] Matsuda, M., Two methods of integrating Monge-Ampère’s equations. II, Trans. Amer. Math. Soc., 166 (1972), 371386.
[9] Matsuda, M., Integration of equations of Imschenetsky type by integrable systems, Proc. Japan Acad., 47 (Suppl. II) (1971), 965969.
[10] Matsuda, M., Reduction of Monge-Ampère’s equations by Imschenetsky transformations, J. Math. Soc. Japan, 25 (1973), 4370.
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# Bäcklund transformations of the first kind associated with Monge-Ampère’s equations

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