On the location of poles for the Ablowitz-Segur family of solutions to the second Painlev\'e equation

TL;DR

This paper demonstrates that solutions in the Ablowitz-Segur family of the second Painlevé equation are free of poles in a specific complex plane region, using operator-norm estimates and numerical illustrations.

Contribution

It introduces a simple operator-norm estimate to identify pole-free regions for these solutions, advancing understanding of their complex behavior.

Findings

Solutions are pole-free in a well-defined complex region.

Operator-norm estimates effectively identify pole-free zones.

Numerical examples support the theoretical results.

Abstract

Using a simple operator-norm estimate we show that the solution to the second Painlev\'e equation within the Ablowitz-Segur family is pole-free in a well defined region of the complex plane of the independent variable. The result is illustrated with several numerical examples.