Asymptotic Studies on Orthogonal Polynomials and Painlevé Transcendents
對正交多項式和潘勒維方程的漸進研究
Student thesis: Doctoral Thesis
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Detail(s)
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| Award date | 22 Aug 2018 |
Link(s)
| Permanent Link | https://scholars.cityu.edu.hk/en/theses/theses(3cb7ddee-abee-4aeb-94e6-7c6db533af67).html |
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| Other link(s) | Links |
Abstract
In this thesis, we applied various methods to study asymptotic behaviors of orthogonal polynomials and the Painlevé transcendents.
First, we study a family of orthogonal polynomials φn(z) arising from nonlinear coherent states in quantum optics. We obtain a uniform asymptotics of φn(z) as the polynomial degree n tends to +∞, by using the Wang and Wong’s difference equation method. 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. In addition, the limiting zero distribution of the polynomials φn(z) is provided.
Then, we turn to the study of the inhomogeneous Painlevé II equation
u''(x) = 2u^3(x) + xu(x) − α
where α is a nonzero constant. Using the Deift and 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 show that the real Ablowitz-Segur solutions have no real poles when α ∈ (−1/2, 1/2) and the imaginary Ablowitz-Segur solutions are always pole free on the real line for α ∈ iR.
We also show that the inhomogeneous Painlevé II equation has a family of solutions possessing infinitely many poles on the negative real axis for all α ∈ R. We derive the singular asymptotics when x → −∞ and the corresponding connection formulas by using the Deift and Zhou nonlinear steepest descent method as well.
Finally, we consider the quasi-Ablowitz-Segur and quasi-Hastings-McLeod solutions of the inhomogeneous Painlevé II equation. These 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. Comput. Math., 14 (2014), no. 5, 985–1016).
First, we study a family of orthogonal polynomials φn(z) arising from nonlinear coherent states in quantum optics. We obtain a uniform asymptotics of φn(z) as the polynomial degree n tends to +∞, by using the Wang and Wong’s difference equation method. 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. In addition, the limiting zero distribution of the polynomials φn(z) is provided.
Then, we turn to the study of the inhomogeneous Painlevé II equation
u''(x) = 2u^3(x) + xu(x) − α
where α is a nonzero constant. Using the Deift and 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 show that the real Ablowitz-Segur solutions have no real poles when α ∈ (−1/2, 1/2) and the imaginary Ablowitz-Segur solutions are always pole free on the real line for α ∈ iR.
We also show that the inhomogeneous Painlevé II equation has a family of solutions possessing infinitely many poles on the negative real axis for all α ∈ R. We derive the singular asymptotics when x → −∞ and the corresponding connection formulas by using the Deift and Zhou nonlinear steepest descent method as well.
Finally, we consider the quasi-Ablowitz-Segur and quasi-Hastings-McLeod solutions of the inhomogeneous Painlevé II equation. These 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. Comput. Math., 14 (2014), no. 5, 985–1016).
- Asymptotic Analysis, Orthogonal Polynomials, Painlevé equations
