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Painlevé Transcendents and PT-Symmetric Hamiltonians

Feb 13, 2015
15 pages
Published in:
  • J.Phys.A 48 (2015) 47, 475202
  • Published: Oct 28, 2015
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Abstract: (IOP)
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,y(0)=0, there is a discrete set of initial slopes y′(0)=bny^{\prime} (0)={b}_{n} 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)=cny(0)={c}_{n} that give rise to separatrix solutions. For Painlevé I the large-n asymptotic behavior of b ( )n( ) is bn∼BIn3/5{b}_{n}\sim {B}_{{\rm{I}}}{n}^{3/5} (n→∞n\to \infty ) and that of c ( )n( ) is cn∼CIn2/5{c}_{n}\sim {C}_{{\rm{I}}}{n}^{2/5} (n→∞n\to \infty ), and for Painlevé II the large-n asymptotic behavior of b ( )n( ) is bn∼BIIn2/3{b}_{n}\sim {B}_{{\rm{II}}}{n}^{2/3} (n→∞n\to \infty ) and that of c ( )n( ) is cn∼CIIn1/3{c}_{n}\sim {C}_{{\rm{II}}}{n}^{1/3} (n→∞n\to \infty ). The constants BI,{B}_{{\rm{I}}}, CI,{C}_{{\rm{I}}}, BII,{B}_{{\rm{II}}}, and CII,{C}_{{\rm{II}}}, which are the coefficients in these asymptotic behaviors, are first determined numerically. Then, by using asymptotic methods, they are found analytically by reducing the nonlinear equations to the linear eigenvalue problems associated with the cubic and quartic PT{\mathcal{P}}{\mathcal{T}}-symmetric Hamiltonians H=12p2+2ix3H=\frac{1}{2}{p}^{2}+2{\rm{i}}{x}^{3} and H=12p2−12x4.H=\frac{1}{2}{p}^{2}-\frac{1}{2}{x}^{4}.
Note:
  • 14 pages, 15 figures
  • semiclassical
  • WKB
  • asymptotic
  • eigenvalue
  • separatrix
  • asymptotic behavior
  • PT symmetry
  • Painleve equation
  • numerical calculations
  • energy spectrum