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On the properties of solutions of a system of two nonlinear differential equations associated with the Josephson model

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Abstract

We investigate the analytic properties of solutions of a system of two first-order nonlinear differential equations with an arbitrary parameter \(l\) associated with an overdamped Josephson model. We reduce this system to a system of differential equations that is equivalent to the fifth Painlevé equation with the sets of parameters

$$\biggl(\frac{(1-l)^2}{8}, -\frac{(1-l)^2}{8},0,-2\biggr), \; \biggl(\frac{l^2}{8}, -\frac{l^2}{8},0,-2\biggr).$$

We show that the solution of the third Painlevé equation with the parameters \((-2l, 2l-2,1,-1)\) can be represented as the ratio of two linear fractional transformations of the solutions of the fifth Painlevé equation (with the parameters in the above sequence) connected by a Bäcklund transformation.

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Funding

This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

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  1. Belarusian State University of Informatics and Radioelectronics, Minsk, Belarus

    V. V. Tsegelnik

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  1. V. V. Tsegelnik
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Correspondence to V. V. Tsegelnik.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2024, Vol. 219, pp. 12–16 https://doi.org/10.4213/tmf10642.

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Tsegelnik, V.V. On the properties of solutions of a system of two nonlinear differential equations associated with the Josephson model. Theor Math Phys 219, 539–543 (2024). https://doi.org/10.1134/S0040577924040020

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