On modified NLS, Kaup and NLBq equations: differential transformations and bilinearization

New transformations between the nonlinear Schrödinger, Kaup and non-local Boussinesq equations as well as their modified counterparts are found and analysed. The bilinear representations of these equations, including an alternative bilinear form of the Chen - Lee - Liu equation, are obtained by a direct method based on the Bell's exponential polynomials. Explicit Wronskian solutions to these equations are also presented.