Quasi-integrable non-linear Schrödinger models, infinite towers of exactly conserved charges and bright solitons

Abstract

Deformations of the focusing non-linear Schrödinger model (NLS) are considered in the context of the quasi-integrability concept. We strengthen the results of JHEP 09 (2012) 103 for bright soliton collisions. We addressed the focusing NLS as a complement to the one in JHEP 03 (2016) 005, in which the modified defocusing NLS models with dark solitons were shown to exhibit an infinite tower of exactly conserved charges. We show, by means of analytical and numerical methods, that for certain two-bright-soliton solutions, in which the modulus and phase of the complex modified NLS field exhibit even parities under a space-reflection symmetry, the first four and the sequence of even order charges are exactly conserved during the scattering process of the solitons. We perform extensive numerical simulations and consider the bright solitons with deformed potential V = 2η 2+ |ψ| 2 2+ , ∈ R, η < 0. However, for two-soliton field components without definite parity we also show numerically the vanishing of the first non-trivial anomaly and the exact conservation of the relevant charge. So, the parity symmetry seems to be a sufficient but not a necessary condition for the existence of the infinite tower of conserved charges. The model supports elastic scattering of solitons for a wide range of values of the amplitudes and velocities and the set {η, }. Since the NLS equation is ubiquitous, our results may find potential applications in several areas of non-linear science.

Quasi-integrability in deformed sine-Gordon models and infinite towers of conserved charges

“…The deformations of the relativistic integrable SU(N) Toda [9] (the N = 2 case is the sine-Gordon (SG) model in disguise [3,5,10]) and Bullough-Dodd (BD) [6] models have been shown to posses an infinite number of quantities which are not exactly time-independent but are, however, asymptotically conserved. Similar phenomena have been observed in the deformations of the non-relativistic focusing and defocusing non-linear Schrödinger (NLS) model possessing bright and dark solitons [4,7,8]. For earlier observations on related phenomena, such as the elastic scattering of solitons in some non-integrable theories, see e.g.…”

“…The soliton properties are intimately related to the integrability of the relevant mathematical models in which they arise [1,2]. Some deformations of these theories have been shown to possess solitary waves that behave in a scattering process in a similar way to true solitons [3,4,5,6,7,8,9,10]. The deformations of the relativistic integrable SU(N) Toda [9] (the N = 2 case is the sine-Gordon (SG) model in disguise [3,5,10]) and Bullough-Dodd (BD) [6] models have been shown to posses an infinite number of quantities which are not exactly time-independent but are, however, asymptotically conserved.…”

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

“…[16,17] who considered the modified NLS potential of the form V (|ψ| 2 ) 2+ε , with ε being a perturbation parameter, and proved that such models possess an infinite number of quasi-conserved charges. Exact dark and bright soliton configurations of QI NLS system [18,19] have also been obtained, the latter possessing infinite towers of exactly conserved charges, bringing the system back closer to integrability. QI deformation has also been studied in supersymmetric SG models [20].…”

“…However the values coincide with the scattering with the values they had before. Conservation properties of these QI systems are demonstrated mostly via numerical methods [14,15,16,17,18,19,29] for the lower order hierarchical equations. On the other hand nonholonomically deformed systems remain completely integrable [23,24,25,26,27,28].…”

“…We adopt the NLS and DNLS hierarchies for this purpose as their integrability structures are well-documented. Further, the explicit demonstration of QID and NHD are commonly realized for the lower-order members of these two hierarchies [14,17,18,19,26,30]. We try to realize the QID and NHD genealogies of these two hierarchies which, to the best of our knowledge, has not been done before.…”