Residual Symmetry Reductions and Painlevé Solitons

This letter introduces the novel concept of Painlevé solitons—waves arising from the interaction between Painlevé waves and solitons in integrable systems. Painlevé solitons can also be viewed as solitons propagating against a Painlevé wave background, in analogy to the established notion of elliptic solitons, which refers 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.