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On the complete integrability of gradient systems on manifold of the beta family of the first kind

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Abstract

In this article, it is shown that there exists a gradient system that is Hamiltonian and completely integrable on the beta manifold of the first kind with two parameters and whose existence depends on a potential function with duality on the manifold and which is the solution of the Legendre equation. It is shown that, by making a statistical borrowing from the gamma function, the gradient system remains Hamiltonian and completely integrable. It is shown that the gradient system constructed on the first kind of two-parameter beta manifold admits a Lax pair representation. This also makes it possible to prove its complete integrability and to determine its Hamiltonian function.

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Acknowledgements

We gratefully acknowledge all the members of the group of research in Algebra and Geometry of the University of Maroua, and all the members of the Computer Engineering Department at the Higher National School of Polytechnic in Yaounde I. We also thank Aboubakar Teumsa and Djafsia for their fruitful discussions with the first author.

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Author notes

  1. D. Joseph and B. B. Thomas contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Maroua, 814, Maroua, Far-North, Cameroon

    Prosper Rosaire Mama Assandje & Joseph Dongho

  2. Department of Computer Science, Higher National School of Polytechnic, 8390, Yaounde, Center, Cameroon

    Thomas Bouetou Bouetou

Authors
  1. Prosper Rosaire Mama Assandje
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  2. Joseph Dongho
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  3. Thomas Bouetou Bouetou
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Correspondence to Prosper Rosaire Mama Assandje.

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Communicated by Nihat Ay.

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Appendix A. The coefficient of Riemannian metric

Appendix A. The coefficient of Riemannian metric

$$\begin{aligned} A_{1}(\theta )= & {} -\frac{\int _{0}^{1}\log ^{2}(x)e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx }{\int _{0}^{1}e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx}\\ {}{} & {} + \frac{ \left( \int _{0}^{1}\log (x)e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx\right) ^{2}}{\left( \int _{0}^{1}e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx\right) ^{2}};\\ A_{2}(\theta )= & {} -\frac{\int _{0}^{1}\log (x)\log (1-x)e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx}{ \int _{0}^{1}e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx}\\ {}{} & {} +\frac{\left( \int _{0}^{1}\log (x)e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx\right) \left( \int _{0}^{1}\log (1-x)e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx\right) }{\left( \int _{0}^{1}e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx\right) ^{2}};\\ A_{3}(\theta )= & {} -\frac{\int _{0}^{1}\log ^{2}(1-x)e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx }{\int _{0}^{1}e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx}\\ {}{} & {} + \frac{ \left( \int _{0}^{1}\log (1-x)e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx\right) ^{2}}{\left( \int _{0}^{1}e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx\right) ^{2}}. \end{aligned}$$

and

$$\begin{aligned} B_{1}(\theta )= & {} -\frac{\left( \int _{0}^{1}\log ^{2}(1-x)e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx\right) \left( \int _{0}^{1}e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx\right) }{m(\theta )\left( \int _{0}^{1}e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx\right) ^{2}}\\{} & {} \quad +\frac{\left( \int _{0}^{1}\log (1-x)e^{\left( \theta _{1}\log x+\theta _{2}\log \right) } dx\right) ^{2}}{m(\theta )\left( \int _{0}^{1}e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx\right) ^{2}}; \end{aligned}$$
$$\begin{aligned} B_{2}(\theta )= & {} \frac{\left( \int _{0}^{1}\log (x)\log (1-x)e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx\right) \left( \int _{0}^{1}e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx\right) }{m(\theta )\left( \int _{0}^{1}e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx\right) ^{2}}\\{} & {} \quad -\frac{\left( \int _{0}^{1}\log (x)e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx\right) \left( \int _{0}^{1}\log (1-x)e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx\right) }{m(\theta )\left( \int _{0}^{1}e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx\right) ^{2}}; \end{aligned}$$
$$\begin{aligned} B_{3}(\theta )= & {} -\frac{\left( \int _{0}^{1}\log ^{2}(x)e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx\right) \left( \int _{0}^{1}e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx\right) }{m(\theta )\left( \int _{0}^{1}e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx\right) ^{2}}\\{} & {} \quad -\frac{\left( \int _{0}^{1}\log (x)e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx\right) ^{2}}{m(\theta )\left( \int _{0}^{1}e^{\left( \theta _{1}\log x+\theta _{2}\log (1-x)\right) } dx\right) ^{2}}. \end{aligned}$$

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Mama Assandje, P.R., Dongho, J. & Bouetou Bouetou, T. On the complete integrability of gradient systems on manifold of the beta family of the first kind. Info. Geo. (2024). https://doi.org/10.1007/s41884-023-00130-z

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