LINEARITY INSIDE NONLINEARITY - EXACT-SOLUTIONS TO THE COMPLEX GINZBURG-LANDAU EQUATION

Systems of two linear partial differential equations (PDEs) with constant coefficients define a natural basis of elementary functions to build some exact solutions to nonlinear PDEs. For the one-dimensional complex Ginzburg-Landau equation, the usual representations of the complex field (Re A,Im A) or (A,($) over bar A) are multivalued, which makes difficult the search for exact solutions in this manner. Fortunately, the Painleve analysis naturally introduces an elementary single valued-like representation of A by two fields (Z,grad Theta), respectively complex and real, uniquely defined by an explicit expression for Theta and A = Ze(i Theta). Another important feature is the invariance by parity on A, which increases the class of expected solutions. This allows to retrieve quite easily the four famous solutions of Nozaki and Bekki, represented by the constant coefficients of two linear partial differential equations and a finite set of constants.