On the quasi-Ablowitz–Segur and quasi-Hastings–McLeod solutions of the inhomogeneous Painlevé II equation

Abstract

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.

Total integrals of Ablowitz‐Segur solutions for the inhomogeneous Painlevé II equation

“…Furthermore, the numerical computations performed in [19] suggested that the qAS solution determined by the parameters (1.14) has exactly n poles on the real line. Recently, the predictions were rigorously justified in [14], where results concerning the residues of these poles were also obtained. If we put α = 0 and allow the parameter k to be an arbitrary complex number, then we call u(x; 0, k) the complex AS solution for the homogeneous PII equation.…”

Singular asymptotics for solutions of the inhomogeneous Painlevé II equation

“…2]. If one extends the value of α to |α| > 1 2 and keeps |k| < | cos(πα)|, we have proved that the asymptotic behavior of u(x; α) is the same as the AS solutions, but [ |α| + 1 2 ] simple poles will appear on the real axis; see [16,Thm. 1].…”

“…In recent years, the existence and monotonic properties of these HM solutions have been studied in [11,13,36]. See also [16] for the properties of the qHM solutions.…”

Open Problems for Painlevé Equations

“…(iii) For P II with α = 0, there are analogs of the Ablowitz-Segur and Hastings-McLeod solutions, known as the quasi-Ablowitz-Segur solution and the quasi-Hastings-McLeod solution [43,59,60]; see also [49,70,72,134]. There is an extensive literature regarding the asymptotics for P II (1.2) when α = 0.…”

“…see Dai and Hu [59,60]. For the quasi-Hastings-McLeod solution, Claeys, Kuijlaars and Vanlessen [43] show that there exists a unique solution which is pole-free on the real axis with the asymptotic behaviours…”