Residual Symmetry Reductions and Painlev\'e Solitons

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.