A unifying approach to the solution of certain evolution equations is demonstrated by a method of explicit construction of solutions to such equations, which is good enough to provide solutions for given initial values, under certain restrictions. This is done by the explicit solution of inverse scattering problems associated with the Schrödinger operator and the Zakharov-Shabat system, respectively. Moreover, the relation to zero-curvature conditions and Lax pairs is investigated. It is shown that the zero-curvature formulation can be used to describe exactly the class of treated equations.