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A study on Darboux polynomials and their significance in determining other integrability quantifiers: A case study in third-order nonlinear ordinary differential equations

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Abstract

In this paper, we present a method for deriving quantifiers of the extended Prelle–Singer (PS) method using Darboux polynomials for third-order nonlinear ordinary differential equations. By knowing the Darboux polynomials and their co-factors, we extract the extended PS method’s quantities without evaluating the PS method’s determining equations. We consider three different cases of known Darboux polynomials. In the first case, we prove the integrability of the given third-order nonlinear equation by utilising the quantifiers of the PS method from the two known Darboux polynomials. If we know only one Darboux polynomial, then the integrability of the given equation will be dealt as Case 2. Likewise, Case 3 discusses the integrability of the given system where we have two Darboux polynomials and one set of PS method quantity. The established interconnection not only helps in deriving the integrable quantifiers without solving the underlying determining equations, but also provides a way to prove the complete integrability and helps us in deriving the general solution of the given equation. We demonstrate the utility of this procedure with three different examples.

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Acknowledgements

The work of MS forms part of a research project sponsored by National Board for Higher Mathematics (NBHM), Government of India, under Grant No. 02011/20/2018 NBHM(RP)/R &D II/15064. RMS is funded by the Centre for Computational Modeling, Chennai Institute of Technology, India, Vide funding number CIT/CCM/2022/RP-006.

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Authors and Affiliations

  1. Centre for Computational Modeling, Chennai Institute of Technology, Chennai, 600 069, India

    R Mohanasubha

  2. Department of Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli, 620 024, India

    M Senthilvelan

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  1. R Mohanasubha
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  2. M Senthilvelan
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Correspondence to R Mohanasubha.

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Mohanasubha, R., Senthilvelan, M. A study on Darboux polynomials and their significance in determining other integrability quantifiers: A case study in third-order nonlinear ordinary differential equations. Pramana - J Phys 97, 28 (2023). https://doi.org/10.1007/s12043-022-02507-8

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  • DOI: https://doi.org/10.1007/s12043-022-02507-8

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