Meromorphic Solution of the Degenerate Third Painlevé Equation Vanishing at the Origin

Abstract

We prove that there exists the unique odd meromorphic solution of dP3, u(τ ) such that u(0) = 0, and study some of its properties, mainly: the coefficients of its Taylor expansion at the origin and asymptotic behaviour as τ → +∞. Contents 9 Positiveness of Re u(τ ) 51 References 53where , a, b ∈ C. We recall that in the case b = 0 equation (1.1) can be integrated in elementary functions, Otherwise, b = 0, both parameters, and b, can be fixed arbitrarily in C\0 by arXiv:1809.00122v5 [math.CA]

Trans-Series Asymptotics of Solutions to the Degenerate Painlevé III Equation: A Case Study

“…(ii) In [41], an extensive number-theoretic and asymptotic analysis of the universal special monodromic solution considered in [54] is presented: the author studies a particular meromorphic solution of the DP3E (1.1) that vanishes at the origin; more specifically, it is proved that, for −i2a ∈ Z, the aforementioned solution exists and is unique, and, for the case a−i/2 ∈ Z, this solution exists and is unique provided that u(τ ) = −u(−τ ). The bulk of the analysis presented in [41] focuses on the study of the Taylor expansion coefficients of the solution of the DP3E (1.1) that is holomorphic at τ = 0; in particular, upon invoking the 'normalisation condition' b = a and taking ε = +1, it is shown that, for general values of the parameter a, these coefficients are rational functions of a 2 that possess remarkable number-theoretic properties: en route, novel notions such as super-generating functions and quasi-periodic fences are introduced. The author also studies the connection problem for the Suleimanov solution of the DP3E (1.1).…”

“…(iii) Unlike the physical optics context adopted in [54], the authors of [5] provide a colossal Riemann-Hilbert problem (RHP) asymptotic analysis of the solution of the focusing NLSE, i∂ T Ψ+ 1 2 ∂ 2 X Ψ+ |Ψ| 2 Ψ = 0, by considering the rogue wave solution Ψ(X, T) of infinite order, that is, a scaling limit of a sequence of particular solutions of the focusing NLSE modelling so-called rogue waves of ever-increasing amplitude, and show that, in the regime of large variables R 2 ∋ (X, T) when |X| → +∞ in such a way that T|X| −3/2 − 54 −1/2 = O(|X| −1/3 ), the rogue wave of infinite order Ψ(X, T) can be expressed explicitly in terms of a function V(y) extracted from the solution of the Jimbo-Miwa Painlevé II (PII) RHP for parameters p = ln(2)/2π and τ = 1; 4 in particular, Corollary 6 of [5] presents the leading term of the T → +∞ asymptotics of the rogue wave of infinite order Ψ(0, T) (see, also, Theorem 2 and Section 4 of [4]), 5 which, in the context of the DP3E (1.1), coincides, up to a scalar, τ -independent factor, with exp(i φ(τ )), T = τ 2 , where, given the solution, denoted by û(τ ), say, of the DP3E (1.1) studied in [41] for the monodromy data corresponding to a = i/2 (and a suitable choice for the parameter b), φ(τ ) is the general solution of the ODE φ′ (τ ) = 2aτ −1 +b(û(τ )) −1 (for additional information regarding the function φ(τ ), see, for example, Subsection 1.3, Proposition 1.3.1 below).…”

“…Remark 1.1.3. The results of this work, in conjunction with those of [42,43], will be applied in an upcoming series of studies on uniform asymptotics of integrals of solutions to the DP3E (1.1) and related functions: for the monodromy data considered in [41], preliminary τ → +∞ asymptotics for εb > 0 have been presented in [44].…”

Asymptotics of Integrals of Some Functions Related to the Degenerate Third Painlevé Equation

“…It is proved in [7] that for all a ∈ C \ iZ, there exists the unique odd meromorphic solution of Equation (1) such that u(0) = 0. The asymptotic calculation of the integrals for this solution is considered by taking the simplest contour, L(0, τ ) = [0, τ ], τ ∈ R + , τ → +∞.…”

“…It is important to note that in Appendix B of the subsequent paper [2], inconsistencies in the paper [1] were located and rectified. Furthermore, as explained in Section 7 of [7], due to the discrepancy in the definition of the canonical solutions and the corresponding linear ODE, one has to add to the asymptotics of the function ϕ, obtained with the help of the results in [1,2], the term a ln τ .…”