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Darboux formulae for linear hyperbolic equations in the discrete case

Published online by Cambridge University Press:  12 September 2025

Sergey Smirnov
Sergey Smirnov*
Affiliation:
School of Mathematics and Statistics, University of Glasgow, Glasgow, UK
*
Email: Sergey.Smirnov@glasgow.ac.uk
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Abstract

In the second half of the 19th century, Darboux obtained determinant formulae that provide the general solution for a linear hyperbolic second-order PDE with the finite Laplace series. These formulae played an important role in his study of the theory of surfaces and, in particular, in the theory of conjugate nets. During the last three decades, discrete analogues of conjugate nets (Q-nets) were actively studied. Laplace series can be defined also for hyperbolic difference operators. We prove discrete analogues of Darboux formulae for discrete and semi-discrete hyperbolic operators with finite Laplace series.

Keywords

Linear hyperbolic equationLaplace seriesLaplace invariantsDarboux determinant formulae

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Type
Research Article
Information
Glasgow Mathematical Journal , First View , pp. 1 - 18
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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