Painleve Transcendents and PT-Symmetric Hamiltonians

Unstable separatrix solutions for the first and second Painlev'e transcendents are studied both numerically and analytically. For a fixed initial condition, say $y(0)=0$, there is a discrete set of initial slopes $y^{\mathrm{\prime }}(0)=b_n$ that give rise to separatrix solutions. Similarly, for a fixed initial slope, say $y^{\mathrm{\prime }}(0)=0$, there is a discrete set of initial values $y(0)=c_n$ that give rise to separatrix solutions. For Painlev'e I the large-$n$ asymptotic behavior of $b_n$ is $b_n\sim B_{\mathrm{I}}{n}^{3\mathrm{/}5}$ and that of $c_n$ is $c_n\sim C_{\mathrm{I}}{n}^{2\mathrm{/}5}$, and for Painlev'e II the large-$n$ asymptotic behavior of $b_n$ is $b_n\sim B_{\mathrm{I}\mathrm{I}}{n}^{2\mathrm{/}3}$ and that of $c_n$ is $c_n\sim C_{\mathrm{I}\mathrm{I}}{n}^{1\mathrm{/}3}$. The constants $B_{\mathrm{I}}$, $C_{\mathrm{I}}$, $B_{\rm II}$,and$C_{\rm II}$ are first determined numerically. Then, they are found analytically and in closed form by reducing the nonlinear equations to the linear eigenvalue problems associated with the cubic and quartic PT-symmetric Hamiltonians $H=\frac{1}{2}{p}^2+2i{x}^3$ and $H=\frac{1}{2}{p}^2-\frac{1}{2}{x}^4$.