Residual Symmetry Reductions and Painlev\'e Solitons

TL;DR

This paper introduces Painlevé solitons, a new type of wave interaction in integrable systems, constructed via a novel symmetry method, expanding the understanding of soliton dynamics on Painlevé wave backgrounds.

Contribution

It presents the concept of Painlevé solitons and develops a symmetry decomposition method to explicitly construct these solitons for KdV and Boussinesq equations.

Findings

Defined Painlevé solitons as waves interacting with Painlevé backgrounds.

Constructed explicit Painlevé II solitons for KdV.

Constructed explicit Painlevé IV solitons for Boussinesq.

Abstract

This letter introduces the novel concept of Painlev\'e solitons -- waves arising from the interaction between Painlev\'e waves and solitons in integrable systems. Painlev\'e solitons may also be viewed as solitons propagating against a Painlev\'e wave background, in analogy with the established notion of elliptic solitons, which refer to solitons on an elliptic wave background. By employing a novel symmetry decomposition method aided by nonlocal residual symmetries, we explicitly construct (extended) Painlev\'e II solitons for the Korteweg-de Vries (KdV) equation and (extended) Painlev\'e IV solitons for the Boussinesq equation.