The structure of WTC expansions and applications
نویسندگان
چکیده
We construct generalized Painlev e 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 rst author. The results are illustrated on a family of fth-order models for water-waves, and other examples. We also give necessary and su cient conditions for 1 to be a resonance. 127 Vincent Hall, School of Mathematics, 206 Church Street S. E., Minneapolis, MN 55455-0487.
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