Painlevé Analysis, Lie Symmetries and Abundant Wave Solutions for Family Fifth Order Kdv Equations

In this paper, we study integrability, similarity reduction and obtaining abundant solutions for the family fifth-order KdV equation. This equation expresses five different forms of the KdV equation, each of these equations has different applications in many fields, including fluid mechanics, ocean science and optics. We utilized Painlevé property for the governing equation to prove that the equation possesses Painlevé test. Then, the symmetry method is used to study the similarity reductions for the governing equation. Subsequently, we obtained a novel type of exact solutions for family KdV fifth-order by using \((G'/G)\)-expansion method. The obtained solutions included hyperbolic and trigonometric functions. The solutions are also presented in 3D shapes to show their properties contained kink wave, singular wave, anti-kink wave, periodic wave and solitary wave solution.