Painleve Transcendents and PT-Symmetric Hamiltonians

الأصل · EN

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 In³/⁵ and that of cₙ is cₙ C In²/ ⁵, and for Painlevé II the large-n asymptotic behavior of bₙ is bₙ B IIn²/³ and that of cₙ is cₙ C IIn¹/³. The constants B I, C I, B II, and C 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=1/2p²+2ix³ and H=1/2p²-1/2x⁴.