2021/07/11 by Carl M. Bender, Javad Komijani, Bender, Carl M. +1 · 1 citation
Physics and Astronomy · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.2107.04935
openalex publication_date 2021/07/11 · openalex created_date 2022/04/03 · openalex updated_date 2026/08/01
This paper is an addendum to earlier papers \citeR1,R2 in which it was shown that the unstable separatrix solutions for Painlevé I and II are determined by PT-symmetric Hamiltonians. In this paper unstable separatrix solutions of the fourth Painlevé transcendent are studied numerically and analytically. For a fixed initial value, say y(0)=1, a discrete set of initial slopes y'(0)=bn give rise to separatrix solutions. Similarly, for a fixed initial slope, say y'(0)=0, a discrete set of initial values y(0)=cn give rise to separatrix solutions. For Painlevé IV the large-n asymptotic behavior of bn is bn∼ B\rm IVn3/4 and that of cn is cn∼ C\rm IV n1/2. The constants B\rm IV and C\rm 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) p2+(1)/(8) x6.