Nonlinear Ordinary and Partial Differential Equations

The automation of the traditional Painleve ́ test in Mathematica is discussed. The package PainleveTest.m allows for the testing of polynomial systems of nonlinear ordinary and partial differential equations which may be parameterized by arbitrary functions (or constants). Except where limited by memory, there is no restriction on the number of independent or dependent variables. The package is quite robust in determining all the possible dominant behaviors of the Laurent series solutions of the differential equation. The omission of valid dominant behaviors is a common problem in many implementations of the Painleve ́ test, and these omissions often lead to erroneous results. Finally, our package is compared with the other available implementations of the Painleve ́ test.