Traveling Wave Solutions of the Burgers – Gardner Equation with a Nonlinear Source

The Burgers – Gardner equation with a nonlinear source is considered. The Cauchy problem for this equation cannot be solved by the inverse scattering transform and we look for the traveling wave solutions. The Painlevé test is used to find constraints on the parameters of the equation for its integrability. The first integral of the nonlinear ordinary differential equation is obtained taking into account the results of the Painlevé test. The first integral thus found is used to obtain the general solution of the equation. Some exact solutions are found by means of the simplest equation method.