We consider the second Painlevé equationwhere α is a nonzero constant. Using the Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems, we rigorously prove the asymptotics as x → ±∞ for both the real and purely imaginary Ablowitz-Segur solutions, as well as the corresponding connection formulas. We also show that the real AblowitzSegur solutions have no real poles when α ∈ (−1/2, 1/2).2010 Mathematics Subject Classification. Primary 41A60, 33C45.
Abstract. In this paper, we study a family of orthogonal polynomials {φ n (z)} arising from nonlinear coherent states in quantum optics. Based on the three-term recurrence relation only, we obtain a uniform asymptotic expansion of φ n (z) as the polynomial degree n tends to infinity. Our asymptotic results suggest that the weight function associated with the polynomials has an unusual singularity, which has never appeared for orthogonal polynomials in the Askey scheme. Our main technique is the Wang and Wong's difference equation method. In addition, the limiting zero distribution of the polynomials φ n (z) is provided.
We consider the quasi-Ablowitz-Segur and quasi-Hastings-McLeod solutions of the inhomogeneous Painlevé II equationThese solutions are obtained from the classical Ablowitz-Segur and Hastings-McLeod solutions via the Bäcklund transformation, and satisfy the same asymptotic behaviors when x → ±∞. For |α| > 1/2, we show that the quasi-Ablowitz-Segur and quasi-Hastings-McLeod solutions possess [ |α| + 1 2 ] simple poles on the real axis, which rigorously justifies the numerical results in Fornberg and Weideman (Found.
We consider a family of solutions to the Painlevé II equationwhich have infinitely many poles on (−∞, 0). Using Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems, we rigorously derive their singular asymptotics as x → −∞. In the meantime, we extend the existing asymptotic results when x → +∞ from α − 1 2 / ∈ Z to any real α. The connection formulas are also obtained.2010 Mathematics Subject Classification. Primary 41A60, 33C45.
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