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The "Painlevé analysis" is quite often perceived as a collection of tricks reserved to experts. The aim of this course is to demonstrate the contrary and to unveil the simplicity and the beauty of a subject which is in fact the theory of the (explicit) integration of nonlinear differential equations.To achieve our goal, we will not start the exposition with a more or less precise "Painlevé test". On the contrary, we will finish with it, after a gradual introduction to the rich world of singularities of nonlinear differential equations, so as to remove any cooking recipe.The emphasis is put on embedding each method of the test into the well known theorem of perturbations of Poincaré. A summary can be found at the beginning of each chapter.

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A major drawback of most methods to find analytic expressions for solitary waves is the a priori restriction to a given class of expressions. To overcome this difficulty, we present a new method, applicable to a wide class of autonomous equations, which builds as an intermediate information the first order autonomous ODE satisfied by the solitary wave. We discuss its application to the cubic complex one-dimensional Ginzburg-Landau equation, and conclude to the elliptic nature of the yet unknown most general solitary wave.

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Exact solitary wave solutions of the one-dimensional quintic complex Ginzburg-Landau equation are obtained using a method derived from the Painlevé test for integrability. These solutions are expressed in terms of hyperbolic functions, and include the pulses and fronts found by van Saarloos and Hohenberg. We also find previously unknown sources and sinks. The emphasis is put on the systematic character of the method which breaks away from approaches involving somewhat ad hoc Ansätze.