Weak perturbative Painleve analysis and descending series solutions

We consider a weak extension of the recently introduced perturbative Painleve test, whereby as in the weak extension of the standard Painleve test finite branching is allowed. This extension allows us to capture even more information about the singularity structure of the solutions to nonlinear ordinary (and partial) differential equations, providing a means of uncovering logarithmic branching which might otherwise be missed. Such logarithmic branching might be thought of as providing a greater obstacle to integrability than does finite branching alone. When our new test is passed the resulting expansion is a Puiseux series in xi = (x+x(0)), i.e. doubly infinite in rational powers of xi. The above-mentioned perturbative Painleve test is included as a special case. However, we are also interested in finding the most efficient means of uncovering such logarithmic branching; we find that for some examples the way to do this is by constructing a descending series solution. For some equations this means that logarithmic branching is exhibited in a particular solution rather than in the general solution, The construction of descending series solutions can be undertaken as a special case of the perturbative Painteve algorithm. The techniques developed here then allow us to reconsider the analysis of the ordinary differential equation U-xx+UUx+kU(3) = 0.