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The objective of this work is to explore the class of equations of theNon-linear Schrodinger type by employing the Adler-Kostant-Symes theorem andthe Tu methodology.In the first part of the work, the AKS theory is discussedin detail showing how to obtain the non-linear equations starting from asuitably chosen spectral problem.Equations derived by this method includedifferent members of the NLS family like the NLS, the coupled KdV type NLS, thegeneralized NLS, the vector NLS, the Derivative NLS, the Chen-Lee-Liu and theKundu-Eckhaus equations. In the second part of the paper, the steps in the Tumethodology that are used to formulate the hierarchy of non-linear evolutionequations starting from a spectral problem, are outlined. The AKNS,Kaup-Newell, and generalized DNLS hierarchies are obtained by using thisalgorithm. Several reductions of the hierarchies are illustrated. The famoustrace identity is then applied to obtain the Hamiltonian structure of thesehierarchies and establish their complete integrability. In the last part of thepaper, the non-holonomic deformation of the class of integrable systemsbelonging to the NLS family is studied. Equations examined include the NLS,coupled KdV-type NLS and Derivative NLS (both Kaup-Newell and Chen-Lee-Liuequations). NHD is also applied to the hierarchy of equations in the AKNSsystem and the KN system obtained through application of the Tumethodology.Finally, we discuss the connection between the two formalisms andindicate the directions of our future endeavour in this area.

Study of the family of Nonlinear Schrodinger equations by using the  Adler-Kosant-Symes framework and the Tu methodology and their Non-holonomic  deformation