In this talk, a method to construct oscillatory soliton solutions of integrable equations that is both Lax and Hirota integrable is presented, taking the Ablowitz-Ladik equation as an illustrative example. The Baker-Akhiezer functions in the field of algebraic solutions and the tau-functions in the field of the direct method are combined to solve the spectral problems, construct Darboux transformations, and drive exact solutions.