Nonisospectral scattering problems and similarity reductions

• Mappings between nonisospectral and isospectral scattering problems.
• Connection between two methods of deriving Painlevé hierarchies is explored.
• Korteweg–de Vries and dispersive water wave cases are considered.

We give mappings between hierarchies having nonisospectral scattering problems and hierarchies having isospectral scattering problems. Special cases of these changes of variables involve similarity variables, and similarity solutions of a hierarchy are seen to correspond to time-independent solutions of an equivalent hierarchy. We thus explain why the use of nonisospectral scattering problems and similarity reductions yield the same Painlevé hierarchies. As examples we consider the Korteweg–de Vries hierarchy and the dispersive water wave hierarchy.