# Study of Non-Holonomic Deformations of Non-local integrable systems belonging to the Nonlinear Schrodinger family

## TL;DR

This paper investigates non-holonomic deformations of non-local integrable systems in the Nonlinear Schrödinger family, using bi-Hamiltonian and Lax pair methods to derive and analyze these deformations.

## Contribution

It introduces a systematic approach to derive non-holonomic deformations of non-local NLS systems via bi-Hamiltonian structures and Lax pairs, establishing their equivalence.

## Key findings

- Explicit bi-Hamiltonian structures for non-local NLS equations
- Derivation of non-holonomic deformations using Kupershmidt ansatz
- Equivalence between bi-Hamiltonian and Lax pair deformations for certain systems

## Abstract

The non-holonomic deformations of non-local integrable systems belonging to the Nonlinear Schrodinger 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 Schrodinger and Derivative Nonlinear Schrodinger (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