The structure of WTC expansions and applications
Résumé
We construct generalized Painlevé expansions with logarithmic terms
for a general class of (`nonintegrable') scalar equations, and
describe their structure in detail. These expansions were introduced
without logarithms by Weiss-Tabor-Carnevale (WTC). The construction
of the formal solutions is shown to involve semi-invariants of binary
forms, and tools from invariant theory are applied to the
determination of the type of logarithmic terms that are required for
the most general singular series. The structure of the series depends
strongly on whether 1 is or is not a resonance. The convergence of
these series is obtained as a consequence of the general results of
Littman and the first author. The results are illustrated on a family
of fifth-order models for water-waves, and other examples. We also
give necessary and sufficient conditions for $-1$ to be a resonance.