On the location of poles for the Ablowitz-Segur family of solutions to the second Painlevé equation (Q2882663)

The author proves that all solutions to the second Painlevé equation \(u_{ss}=su+2u^3\) satisfying the asymptotic condition \(u(s)\simeq k\, \text{Ai}(s)\), \(s\to+\infty\), \(k\in{\mathbb C}\), including the famous Ablowitz-Segur and Hastings-McLeod solutions, have no one pole in the region \(\text{Re}(s^{3/2})>\tfrac{3}{2}\ln|k|\). The proof is based on the fact that the poles of the second Painlevé transcendent coincide with zeros of the Fredholm determinant \(\det(\text{Id}_{L^2([s,\infty))}-k^2K_{\text{Ai}}|_{[s,\infty)})\) (\(\tau\)-function). Here, \(K_{\text{Ai}}\bigr|_{[s,\infty)}\) is the operator with the Airy kernel acting on the semi-infinite interval \([s,\infty)\). The author observes that, in the Fourier space, this operator appears to be a composition of multiplication and Cauchy operators, and thus its norm is easy to estimate. In the interior of the above indicated region in the \(s\)-complex plane, the author finds that \(\|k^2K_{\text{Ai}}|_{[s,\infty)}\|<1\) and therefore the determinant does not vanish.