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arXiv · 2609.06217

On B\"acklund transformations preserving the Darboux integrability of hyperbolic equations

Sergey Ya. Startsev
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Abstract

This paper deals with two kinds of B\"acklund transformations for scalar hyperbolic partial differential equations. We prove that both these types of transformations map solutions of a Darboux integrable equation into solutions of, generally speaking, another but also Darboux integrable equation. The latter fact can be used to roughly check the completeness of a list of Darboux integrable equations. To illustrate this, we apply the above transformations to several equations from a well-known list of Darboux integrable equations and, as a result, obtain a Darboux integrable equation which is absent in this list, but is already known at present. As a generalization of the last equation, we construct a family of Darboux integrable equations that is parametrized by three arbitrary functions, each of which depends on two arguments. This family is probably new.

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Sergey Ya. Startsev. 2026-09-05. On B\"acklund transformations preserving the Darboux integrability of hyperbolic equations. https://doi.org/10.1134/s1995080223050517

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