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

Abstract

The hierarchy of equations belonging to two different but related integrable systems, the nonlinear Schrödinger (NLS) and its derivative variant the derivative nonlinear Schrödinger (DNLS), is subjected to two distinct deformation procedures, viz., quasi-integrable deformation that generally does not preserve the integrability, with the system only asymptotically integrable, and non-holonomic deformation that does. Quasi-integrable deformation 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 quasi-integrable anomaly is observed at the level of equations of motion which suggests that at that level, the quasi-integrable deformation may be identified as some integrable deformation. Non-holonomic deformation is applied to the NLS hierarchy generally, with the specific focus on the NLS equation itself and on the coupled Korteweg-de Vries type NLS equation. For the DNLS hierarchy, the Kaup-Newell and Chen-Lee-Liu equations are deformed non-holonomically, and subsequently, different aspects of the results are discussed.