Properties of solutions of two second-order differential equations with the Painlevé property
The author considers a Hamiltonian system equivalent to the Painlevé II equation with respect to one component and to the Painlevé XXXIV equation with respect to another. He obtains two Bäcklund transformations (direct and inverse) for the solutions of the Painlevé XXXIV equation. Based on this, a nonlinear functional relation for the solutions of the Painlevé XXXIV equation with different values of its parameter is obtained. Further, he constructs a second-degree and second-order nonlinear differential equation with an arbitrary analytic function \(F(t)\) and an arbitrary parameter $\gamma$ that is a Painlevé-type equation, which for $\gamma=1$ is the canonical equation XXVII in the Ince list in the case $m=2$. Precisely, he obtains a Painlevé-type equation that reduces to the above mentioned equation for $F(t)=-t$ and $\gamma=0$. He finally shows that the direct and inverse Bäcklund transformations coincide with the pair of Bäcklund transformations for the Painlevé XXXIV equation.
scientific article; zbMATH DE number 7364270