This short survey presents the essential features of what is called Painleve analysis, i.e. the set of methods based on the singularities of differential equations in order to perform their explicit integration. Full details can be found in textit{Th
e Painleve handbook} or in various lecture notes posted on arXiv.
In evolution equations for a complex amplitude, the phase obeys a much more intricate equation than the amplitude. Nevertheless, general methods should be applicable to both variables. On the example of the traveling wave reduction of the complex cub
ic-quintic Ginzburg-Landau equation (CGL5), we explain how to overcome the difficulties arising in two such methods: (i) the criterium that the sum of residues of an elliptic solution should be zero, (ii) the construction of a first order differential equation admitting the given equation as a differential consequence (subequation method).
We show that all meromorphic solutions of the stationary reduction of the real cubic Swift-Hohenberg equation are elliptic or degenerate elliptic. We then obtain them all explicitly by the subequation method, and one of them appears to be a new elliptic solution.
In the system made of Korteweg-de Vries with one source, we first show by applying the Painleve test that the two components of the source must have the same potential. We then explain the natural introduction of an additional term in the potential o
f the source equations while preserving the existence of a Lax pair. This allows us to prove the identity between the travelling wave reduction and one of the three integrable cases of the cubic Henon-Heiles Hamiltonian system.
The dynamical degenerate four-wave mixing is studied analytically in detail. By removing the unessential freedom, we first characterize this system by a lower-dimensional closed subsystem of a deformed Maxwell-Bloch type, involving only three physica
l variables: the intensity pattern, the dynamical grating amplitude, the relative net gain. We then classify by the Painleve test all the cases when singlevalued solutions may exist, according to the two essential parameters of the system: the real relaxation time tau, the complex response constant gamma. In addition to the stationary case, the only two integrable cases occur for a purely nonlocal response (Real(gamma)=0), these are the complex unpumped Maxwell-Bloch system and another one, which is explicitly integrated with elliptic functions. For a generic response (Re(gamma) not=0), we display strong similarities with the cubic complex Ginzburg-Landau equation.
In order to later find explicit analytic solutions, we investigate the singularity structure of a fundamental model of nonlinear optics, the four-wave mixing model in one space variable z. This structure is quite similar, and this is not a surprise,
to that of the cubic complex Ginzburg-Landau equation. The main result is that, in order to be single valued, time-dependent solutions should depend on the space-time coordinates through the reduced variable xi=sqrt{z} exp(-t / tau), in which tau is the relaxation time.
