Exact solutions to a system of ODEs arising in classification of diagonalizable hydrodynamic chains
Abstract
We obtain exact solutions of a system of two third--order ordinary differential equations arising in the classification of diagonalizable hydrodynamic chains. We show that the system admits a four--dimensional solvable Lie algebra of point symmetries. Although this algebra is not large enough for the complete integrability by quadratures, it enables a reduction to a two--dimensional first--order system. A further symmetry reduction leads to an Abel equation whose implicit solutions define a family of algebraic curves. By analysing their singular cases and deriving rational parametrizations, we obtain the solutions to the reduced system in parametric form. The solutions to the original third--order system are then reconstructed by four successive quadratures using Lie--Bianchi's theorem. The resulting families contain a previously known specific solution as a particular case and provide new diagonalizable hydrodynamic chains.
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BibTeX
@article{flor2026exact,
title = {{Exact solutions to a system of ODEs arising in classification of diagonalizable hydrodynamic chains}},
author = {Beltrán de la Flor and Adrián Ruiz and Concepción Muriel},
journal = {Open Communications in Nonlinear Mathematical Physics},
year = {2026},
month = oct,
doi = {10.46298/ocnmp.18629},
url = {https://hal.science/hal-05670162v2},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - JOUR
TI - Exact solutions to a system of ODEs arising in classification of diagonalizable hydrodynamic chains
AU - Beltrán de la Flor
AU - Adrián Ruiz
AU - Concepción Muriel
PY - 2026
DA - 2026/10/27
JO - Open Communications in Nonlinear Mathematical Physics
DO - 10.46298/ocnmp.18629
UR - https://hal.science/hal-05670162v2
AB - We obtain exact solutions of a system of two third--order ordinary differential equations arising in the classification of diagonalizable hydrodynamic chains. We show that the system admits a four--dimensional solvable Lie algebra of point symmetries. Although this algebra is not large enough for the complete integrability by quadratures, it enables a reduction to a two--dimensional first--order system. A further symmetry reduction leads to an Abel equation whose implicit solutions define a family of algebraic curves. By analysing their singular cases and deriving rational parametrizations, we obtain the solutions to the reduced system in parametric form. The solutions to the original third--order system are then reconstructed by four successive quadratures using Lie--Bianchi's theorem. The resulting families contain a previously known specific solution as a particular case and provide new diagonalizable hydrodynamic chains.
SN - 2802-9356
ER -