2015/02/13 by Carl M. Bender, Javad Komijani, Bender, Carl M. +1 · 2 citations
Physics and Astronomy · #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.1502.04089
openalex publication_date 2015/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Unstable separatrix solutions for the first and second Painlevé 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'(0)=bn that give rise to separatrix solutions. Similarly, for a fixed initial slope, say y'(0)= 0, there is a discrete set of initial values y(0)=cn that give rise to separatrix solutions. For Painlevé I the large-n asymptotic behavior of bn is bn∼ B\rm In3/5 and that of cn is cn∼ C\rm In2/ 5, and for Painlevé II the large-n asymptotic behavior of bn is bn ∼ B\rm IIn2/3 and that of cn is cn∼ C\rm IIn1/3. The constants B\rm I, C\rm 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=(1)/(2)p2+2ix3 and H=(1)/(2)p2-(1)/(2)x4.