Residual Symmetry Reductions and Painlevé Solitons

2025/11/12 by Li, Yan, Xia, Ya-Rong, Yao, Ruo-Xia +1
#Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Physics (math-ph) #Pattern Formation and Solitons (nlin.PS)

paper · doi:10.48550/arxiv.2511.09077

Abstract

This letter introduces the novel concept of Painlevé solitons -- waves arising from the interaction between Painlevé waves and solitons in integrable systems. Painlevé solitons may also be viewed as solitons propagating against a Painlevé 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) Painlevé II solitons for the Korteweg-de Vries (KdV) equation and (extended) Painlevé IV solitons for the Boussinesq equation.

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