A Study of Nonholonomic Deformations of Nonlocal Integrable Systems Belonging to the Nonlinear Schrödinger Family

Abstract

The non-holonomic deformations of non-local integrable systems belonging to the Nonlinear Schrödinger family are studied using the Bi-Hamiltonian formalism as well as the Lax pair method. The non-local equations are first obtained by symmetry reductions of the variables in the corresponding local systems. The bi-Hamiltonian structures of these equations are explicitly derived. The bi-Hamiltonian structures are used to obtain the non-holonomic deformation following the Kupershmidt ansatz. Further, the same deformation is studied using the Lax pair approach and several properties of the deformation discussed. The process is carried out for coupled non-local Nonlinear Schrödinger and Derivative Nonlinear Schrödinger (Kaup Newell) equations. In case of the former, an exact equivalence between the deformations obtained through the bi-Hamiltonian and Lax pair formalisms is indicated.