The Aspe-Depassier Equation with a Nonlinear Source: Reduction, Painlevé test, First Integrals and General Solutions

A nonlinear partial differential equation describing waves in a convecting fluid with a nonlinear source is considered. Since the Cauchy problem is not solvable by the inverse scattering transform a traveling wave variables is employed to reduce the partial differential equation to fourth-order ordinary differential equation. The Painlevé test is used to derive the parameter constraints of the equation for its integrability. The relationship between the Fuchs indices obtained from the Painlevé test and the form of the first integrals is discussed. Guided by this information the first integrals are constructed. Under four parameter constraints obtained during Painlevé test general solution with four arbitrary constants are found. All such general solutions are expressed in terms of the transcendents of the first Painlev e equation, which are non-classical functions not reducible to known elementary functions.