An explicit description of the connection formula for the first Painlevé equation (Q2744102)

The author considers the first Painlevé equation \({d^2\lambda \over dt^2},= \eta^2(6\lambda^2+t)\), where \(\eta\) is a large parameter. So, at the limit the order is reduced. By using the asymptotic expansion \(\lambda^{(0)}_1(t)= \lambda_0(t)+ \eta^{-1}\lambda_1 (t)+\dots\). and a recursive solution, all the coefficients are found. Using the two parameter solution and using only the homogeneous differential equation, the author obtains an explicit form of the connection formula, and the original equation is reduced to the Hamiltonian system NEWLINE\[NEWLINE{d\lambda\over dt}=\eta{_\partial K_1 \over\partial\nu}, \quad {dv\over dt}= -\eta{_\partial K_1\over \partial\lambda},NEWLINE\]NEWLINE with \(K_1=v^2/2-(2\lambda^3 +t \lambda)\). The author cleverly obtains all Stokes multipliers corresponding to the Stokes regions.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00055].