Geodesic
On B\"acklund transformations for some second-order nonlinear differential equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 217 (2023) no. 2, pp. 391-403

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We obtain second-order nonlinear differential equations (and the associated Bäcklund transformations) with an arbitrary analytic function of the independent variable. These equations (which are not of Painlevé type in general) under certain constraints imposed on an arbitrary analytic function can be reduced, in particular, to the second, third or fourth Painlevé equation. We consider the properties of the Bäcklund transformations for the second-order nonlinear differential equations generated by two systems of two first-order nonlinear differential equations with quadratic nonlinearities in derivatives of the unknown functions.
Keywords: Painlevé property, Painlevé equations, direct and inverse Bäcklund transformations. 
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     author = {V. V. Tsegel'nik},
     title = {On {B\"acklund} transformations for some second-order nonlinear differential equations},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     publisher = {mathdoc},
     volume = {217},
     number = {2},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2023_217_2_a9/}
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V. V. Tsegel'nik. On B\"acklund transformations for some second-order nonlinear differential equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 217 (2023) no. 2, pp. 391-403. http://geodesic.mathdoc.fr/item/TMF_2023_217_2_a9/