Study of quasi-integrable and non-holonomic deformation of equations in the NLS and DNLS hierarchy

TL;DR

This paper investigates quasi-integrable and non-holonomic deformations of the NLS and DNLS hierarchies, revealing that QID preserves asymptotic integrability while NHD maintains integrability, with detailed analysis on various related equations.

Contribution

It introduces and compares quasi-integrable and non-holonomic deformation procedures applied to NLS and DNLS hierarchies, providing new insights into their integrability properties.

Findings

No QI anomaly observed at EOM level, indicating potential integrability.

QID preserves asymptotic integrability without full integrability.

NHD maintains the integrability of the deformed equations.

Abstract

The hierarchy of equations belonging to two different but related integrable systems, the Nonlinear Schr\"odinger and its derivative variant, DNLS are subjected to two distinct deformation procedures, viz. quasi-integrable deformation (QID) that generally do not reserve the integrability, only asymptotically integrable, and non-holonomic deformation (N HD) that does. QID is carried out generically for the NLS hierarchy while for the DNLS hierarchy, it is first done on the Kaup-Newell system followed by other members of the family. No QI anomaly is observed at the level of EOMs which suggests that at that level the QID may be identified as some integrable deformation. NHD is applied to the NLS hierarchy generally as well as with the specific focus on the NLS equation itself and the coupled KdV type NLS equation. For the DNLS hierarchy, the Kaup-Newell(KN) and Chen-Lee-Liu (CLL) equations are deformed non-holonomically and subsequently, different aspects of the results are discussed.