On the integrability of hyperbolic systems of riccati-type equations |
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Authors: | A A Bormisov E S Gudkova F Kh Mukminov |
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Institution: | (1) Sterlitamak State Pedagogical Institute, Sterlitamak, Russia |
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Abstract: | We consider equations of the form Uxy = U * Ux, where U(x, y) is a function taking values in an arbitrary finite-dimensional algebra T over the field ℂ. We show that every
such equation can be naturally associated with two characteristic Lie algebras, Lx and Ly. We define the notion of a ℤ-graded Lie algebraB corresponding to a given equation. We prove that for every equation under consideration, the corresponding algebraB can be taken as a direct sum of the vector spaces Lx and Ly if we define the commutators of the elements from Lx and Ly by means of the zero-curvature relations. Assuming that the algebra T has no left ideals, we classify the equations of the
specified type associated with the finite-dimensional characteristic Lie algebras Lx and Ly. All of these equations are Darboux-integrable.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 2, pp. 261–275, November, 1997. |
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