Geodesic
On the~properties of solutions of a~system of two nonlinear differential equations associated with the~Josephson model
Teoretičeskaâ i matematičeskaâ fizika, Tome 219 (2024) no. 1, pp. 12-16

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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.
Keywords: third Painlevé equation, fifth Painlevé equation, Bäcklund transformation, Josephson model. 
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     author = {V. V. Tsegel'nik},
     title = {On the~properties of solutions of a~system of two nonlinear differential equations associated with {the~Josephson} model},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     number = {1},
     year = {2024},
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     url = {http://geodesic.mathdoc.fr/item/TMF_2024_219_1_a1/}
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V. V. Tsegel'nik. On the~properties of solutions of a~system of two nonlinear differential equations associated with the~Josephson model. Teoretičeskaâ i matematičeskaâ fizika, Tome 219 (2024) no. 1, pp. 12-16. http://geodesic.mathdoc.fr/item/TMF_2024_219_1_a1/