Double Wronskian Solutions of the AKNS and the Classical Boussinesq Hierarchies

The elementary and simple method of Freeman and Nimmo for verification of solutions of the bilinear soliton equations is developed to the Ablowitz-Kaup-Newell-Segur (AKNS) and the classical Boussinesq hierarchies in the bilinear form. Solutions in terms of double Wronskian determinants are obtained. Several determinantal identities which can be derived from the Laplace expansion theorem are utilized in the proof.