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 and its derivative variant, DNLS are subjected to two distinct deformation procedures, viz. quasi-integrable deformation (QID) that generally do not preserve 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. *

Analysis and comparative study of non-holonomic and quasi-integrable deformations of the nonlinear Schrödinger equation

“…3 in analogy with the sine-Gordon system [18]. To our comfort, the B 0 -dependent contribution to the Lax pair exclusively belongs to the Kernel subspace of the corresponding sl(2) loop algebra (to be explicated later) in M which is a signature of a large class of quasi-deformed systems [21,26]. Following this analogy, B 0 itself corresponds to the sine-Gordon potential, V SG (ψ) = 1 16 (1 − cos ψ) and thus, the quasi-sine-Gordon system should correspond to an exclusive deformation B 0 → B when A 0 → A is real, which amounts to [18]:…”

“…It is due to such construction that the system can retain its integrability in spite of being subjected to perturbation. Since the AB system has diverse physical applications it is viable to seek its NHD that potentially can connect the original system to other integrable systems [13,14,15,16,26] with possible physical realizations [29]. This could enhance the collective understanding of such nonlinear systems with particular solution sectors being identified.…”

“…Such a requirement implies that the deformed model contains locally non-conserved charges, along with the conserved ones, with the prior becoming conserved asymptotically. Quasi-integrable deformation (QID) of integrable models particularly satisfy these requirements, as seen in the case of sine-Gordon [20] and NLS [21] systems in the recent past, along with their generalizations [22,26]. They further possess soliton-like configurations similar to solitons of known integrable models [23,24], such as the baby Skyrme models with many potentials and the Ward modified chiral models [19].…”

Shifted nonlocal Kundu type equations: Soliton solutions

“…To find two-soliton solution of the system (1) and ( 2) for β = − 1 4 , 0, and β = − 1 2 , we take N = 2 in ( 21) and ( 22) that is g (1) = e θ 1 + e θ 2 and h (1) = e η 1 + e η 2 , where θ j = k j x + ω j t + α j and η j = l j x + m j t + δ j for j = 1, 2. When we substitute the expansions (20) into the Hirota bilinear forms (10)-( 13) and ( 15)- (18) with that choice of g (1) and h (1) we obtain the functions g, h, f and s as g = g (1) + 3 g (3) , h = h (1) + 3 h (3) , f = 1 + 2 f (2) + 4 f (4) , s = 1 + 2 s (2) + 4 s (4) . (31) Now let us first consider two-soliton solutions of CLL and GI systems, and then KN system.…”

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

“…A comparative study encompassing two different types of deformations, viz. non-holonomic and quasi-integrable, of equations in the NLS and DNLS hierarchy was undertaken in [32].…”