## On the location of poles for the Ablowitz-Segur family of solutions to the second Painlevé equation.(English)Zbl 1254.34123

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.

### MSC:

 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies 34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms 33E17 Painlevé-type functions
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