Rogue-wave and lump patterns associated with the third Painlevé equation

Abstract

We report rogue-wave and lump patterns associated with Umemura polynomials, which arise in rational solutions of the third Painlevé equation. We first show that in many integrable equations such as the nonlinear Schrödinger equation and the Boussinesq equation, when internal parameters of their rogue wave solutions are large and of certain form, then their rogue patterns in the spatial-temporal plane can be asymptotically predicted by root distributions of Umemura polynomials (or equivalently, pole distributions of rational solutions to the third Painlevé equation). Specifically, every simple root of the Umemura polynomial would induce a fundamental rogue wave whose spatial-temporal location is linearly related to that simple root, while a multiple root of the Umemura polynomial would induce a non-fundamental rogue wave in the $O(1)$ neighborhood of the spatial-temporal origin. Next, we show that in a certain class of higher-order lump solutions of the Kadomtsev-Petviashvili-I (KPI) equation, when their internal parameters are large and of certain form, then their lump patterns at $O(1)$ time can also be predicted asymptotically by root distributions of Umemura polynomials, where simple and multiple roots of the polynomial would give rise to fundamental and non-fundamental lumps in the spatial plane, respectively. These results reveal the importance of the third Painlevé equation in studies of nonlinear wave patterns. We also report a new transformation which turns bilinear rogue-wave solutions of the nonlinear Schrödinger equation to higher-order lump solutions of the KPI equation.