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

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

“…In addition, in these last references the authors have found, through numerical simulations, that the first non-trivial anomaly vanishes even in the cases where the space-time parity and spaceparity arguments do not indicate they should vanish. We believe that the results in [7,8] and the ones in the present paper open the way for new investigations on the nature of the quasiintegrability phenomena.…”

“…By considering linear combinations of the asymptotically conserved charges of [3] we show, through analytical and numerical methods, that one tower of them becomes exactly conserved and the other one remains quasi-conserved after the combination; in this way, we have strengthened the arguments of [3] for two-soliton configurations with definite parity. Notice that we have dealt with deformations of the sine-Gordon integrable relativistic field theory and reproduced analogous results to the ones in the non-relativistic non-linear Schrödinger model, as presented in the recent contributions for defocusing NLS with dark solitons [7] and the focusing NLS with bright solitons [8]. In addition, in these last references the authors have found, through numerical simulations, that the first non-trivial anomaly vanishes even in the cases where the space-time parity and spaceparity arguments do not indicate they should vanish.…”

“…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.…”

“…Recently, certain modified defocusing and focusing NLS models [7,8], with dark and bright soliton solutions, respectively, have been shown to exhibit the new feature of an infinite tower of exactly conserved charges. For the special case of two dark (bright) soliton solutions, where the field components are eigenstates of a space-reflection symmetry, they exhibit an alternating sequence of exactly conserved and asymptotically conserved charges for the scattering process of the solitons.…”