Addendum: Painlevé transcendents and PT -symmetric Hamiltonians (2015 J. Phys. A: Math. Theor. 48 475202)

Abstract

Abstract This paper is an Addendum to reference Bender and Komijani (2015 J. Phys. A: Math. Theor. 48 475202) (which stems from an earlier paper Bender et al (2014 J. Phys. A: Math. Theor. 47 235204)), where it was demonstrated that unstable separatrix solutions to the Painlevé equations I and II are determined by PT -symmetric Hamiltonians. Here, unstable separatrix solutions of the fourth Painlevé transcendent are studied numerically and analytically. It is shown that for a fixed initial value such as y(0) = 1 a discrete set of initial slopes y′(0) = b n give rise to separatrix solutions. Similarly, for a fixed initial slope such as y′(0) = 0 a discrete set of initial values y(0) = c n give rise to separatrix solutions. For Painlevé IV the large-n asymptotic behavior of b n is b n ∼ B IV n 3/4 and that of c n is c n ∼ C IV n 1/2. The constants B IV and C IV are determined both numerically and analytically. The analytical values of these constants are found by reducing the nonlinear Painlevé IV equation to the linear eigenvalue equation for the sextic PT -symmetric Hamiltonian H = 1 2 p 2 + 1 8 x 6 .