Painlevé test, first integrals and analytical solutions of the Korteweg–de Vries–Burgers equation with a nonlinear source

Abstract

The Korteweg–de Vries–Burgers equation with a nonlinear source is studied. The Cauchy problem for this equation cannot be solved by the inverse scattering transform in the general case. Therefore, the equation is considered taking into account the traveling wave variables. The Painlevé test is applied to the resulting nonlinear ordinary differential equation to investigate its integrability. It is shown that general solutions of the nonlinear ordinary differential equation can be expressed via the Weierstrass elliptic function and the first Painlevé transcendents under certain parameter constraints. The relationship between the Painlevé test and special methods for finding exact solutions of nonlinear differential equations is discussed. Special methods are used to construct analytical solutions with one and two arbitrary constants. Exact solutions with two arbitrary constants expressed in terms of the Weierstrass elliptic function are obtained. Exact solutions with one arbitrary constant of the Korteweg–de Vries–Burgers equation with a nonlinear source are found using the logistic function method. It is demonstrated that the family of equations for which exact solutions are found is significantly expanded by the use of special methods.