  Exact solutions with the initial conditions are presented in the cubic
duffing equation. These exact solutions are expressed in terms of the leaf
function and the trigonometric function. The leaf functions: $r=sleaf_n(t) $ or
$ r=cleaf_n(t)$ satisfy the ordinary differential equation: $
dx^2/dt^2=-nr^{2n-1}$. The second order differential of the leaf function is
equal to $-n$ times the function raised to the $(2n-1)$ power of the leaf
function. By using the Leaf functions, the exact solutions of the cubic Duffing
equation can be derived under several conditions. In this paper, seven types of
the exact solutions are presented based on the leaf functions. This paper
reveals the derivation of the seven exact solutions. Finally, the waves based
on the exact solutions are visualized by the numerical results in the graphs
