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On the Complete Integrability of the Raychaudhuri Differential System in \(\mathbb {R}^4\) and of a CRNT Model in \(\mathbb {R}^5\)

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Abstract

We study the Darboux integrability of two differential systems with parameters: the Raychaudhuri equation (a relativistic model in \(\mathbb {R}^4\)) and a chemical reaction model in \(\mathbb {R}^5\). We prove that the first one is completely integrable and that the first integrals are of Darboux type. This is the first four-dimensional realistic non-trivial model which is completely integrable with first integrals of Darboux type and for which for a full Lebesgue measure set of the values of the parameters the three linearly independent first integrals are rational. For the second one, we find all its Darboux polynomials and exponential factors and we prove that it is not Darboux integrable.

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Acknowledgements

A. Ferragut is partially supported by the Spanish Government Grant MTM2013-40998-P and by the Universitat Jaume I Grant P1-1B2015-16. C.Valls is supported by Portuguese National Funds through FCT - Fundação para a Ciência e a Tecnologia within the Project PTDC/MAT/117106/2010 and by CAMGSD.

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Authors and Affiliations

  1. Departament de Matemàtiques, Institut Universitari de Matemàtiques i Aplicacions de Castelló (IMAC), Universitat Jaume I, Edifici TI (ESTEC), Av. de Vicent Sos Baynat, s/n, Campus del Riu Sec, 12071, Castelló de la Plana, Spain

    Antoni Ferragut

  2. Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    Claudia Valls

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  1. Antoni Ferragut
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Correspondence to Antoni Ferragut.

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Ferragut, A., Valls, C. On the Complete Integrability of the Raychaudhuri Differential System in \(\mathbb {R}^4\) and of a CRNT Model in \(\mathbb {R}^5\) . Qual. Theory Dyn. Syst. 17, 291–307 (2018). https://doi.org/10.1007/s12346-017-0230-7

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  • DOI: https://doi.org/10.1007/s12346-017-0230-7

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