2017/06/27 by Robert Buckingham, Buckingham, Robert
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics
paper · pdf · doi:10.48550/arxiv.1706.09005
openalex publication_date 2017/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Painleve-IV equation has three families of rational solutions generated\nby the generalized Hermite polynomials. Each family is indexed by two positive\nintegers m and n. These functions have applications to nonlinear wave\nequations, random matrices, fluid dynamics, and quantum mechanics. Numerical\nstudies suggest the zeros and poles form a deformed n by m rectangular grid.\nProperly scaled, the zeros and poles appear to densely fill certain curvilinear\nrectangles as m and n tend to infinity with r=m/n fixed. Generalizing a method\nof Bertola and Bothner used to study rational Painleve-II functions, we express\nthe generalized Hermite rational Painleve-IV functions in terms of certain\northogonal polynomials on the unit circle. Using the Deift-Zhou nonlinear\nsteepest-descent method, we asymptotically analyze the associated\nRiemann-Hilbert problem in the limit as n tends to infinity with m=r*n for r\nfixed. We obtain an explicit characterization of the boundary curve and\ndetermine the leading-order asymptotic expansion of the functions in the\npole-free region.\n