A PERTURBATIVE PAINLEVE APPROACH TO NONLINEAR DIFFERENTIAL-EQUATIONS

We further improve the Painleve test so that negative indices (''reson ances'') can be treated: we demand single valuedness not only for any pole-like expansion as in the usual Painleve test, but also for every solution close to it, represented as a perturbation series in a small parameter epsilon. Order zero is the usual test. Order one, already tr eated in a preliminary paper, reduces to a (linear) Fuchs analysis nea r a regular singularity and allows the introduction of all missing arb itrary coefficients. Higher orders lead to the analysis of a linear, F uchsian type inhomogeneous equation. We obtain an infinite sequence of necessary conditions for the absence of movable logarithmic branch po ints, arising at every integer index, whether positive or negative, an d at every order, those arising at negative indices, including -1, are new, while some conditions may not arise before some high perturbatio n order. We present several illustrative examples. We discuss the unde rstanding of negative indices, and conclude that they are indistinguis hable from positive indices, just as in the Fuchs theory. In particula r, negative indices give rise to doubly infinite Laurent series.