On Bäcklund transformations preserving the Darboux integrability of hyperbolic equations
Abstract
This paper deals with two kinds of Bäcklund 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.
Versions
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BibTeX
@misc{startsev2026bäcklund,
title = {{On Bäcklund transformations preserving the Darboux integrability of hyperbolic equations}},
author = {Sergey Ya. Startsev},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06217},
eprint = {2609.06217},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06217},
}
RIS
TY - UNPB
TI - On Bäcklund transformations preserving the Darboux integrability of hyperbolic equations
AU - Sergey Ya. Startsev
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.06217
UR - https://arxiv.org/abs/2609.06217
AB - This paper deals with two kinds of Bäcklund 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.
ER -