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

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 Painleve equations I and II are determined by PT-symmetric Hamiltonians. Here, unstable separatrix solutions of the fourth Painleve 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 Painleve IV the large-n asymptotic behavior of b(n) is b(n) similar to B(IV)n(3/4) and that of c(n) is c(n) similar to 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 Painleve IV equation to the linear eigenvalue equation for the sextic PT-symmetric Hamiltonian H = 1/2p(2) + 1/8x(6).