Painleve analysis of the non-linear Schrodinger family of equations

The authors apply the Painleve test to the generalised derivative non-linear Schrodinger equation, iu_t=u_xx+iauu*u_x+ibu²u_x+cu³u*² where u* denotes the complex conjugate of u, and a, b and c are real constants, to determine under what conditions the equation might be completely integrable. It is shown that, apart from a trivial multiplicative factor, this equation possesses the Painleve property for partial differential equations as formulated by Weiss, Tabor and Carnevale (1983) only if c=1/4b(2b-a). When then this relation holds, this is equivalent under a gauge transformation to the derivative non-linear Schrodinger equation (DNLS) of Kaup and Newell, which is known to be completely integrable, or else to a linear equation.