Exact N-envelope-soliton solutions of the Hirota equation

We discuss some properties of the soliton equations of the type ∂u/∂t = S[u, ū], where S is a nonlinear operator differential in x, and present the additivity theorems of the class of the soliton equations. On using the theorems, we can construct a new soliton equation through two soliton equations with similar properties. Meanwhile, exact N-envelope-soliton solutions of the Hirota equation are derived through the trace method