Abstract
The concept of quasi-integrability has been examined in the context of deformations of the defocusing non-linear Schrödinger model (NLS). Our results show that the quasi-integrability concept, recently discussed in the context of deformations of the sine-Gordon, Bullough-Dodd and focusing NLS models, holds for the modified defocusing NLS model with dark soliton solutions and it exhibits the new feature of an infinite sequence of alternating conserved and asymptotically conserved charges. For the special case of two dark soliton solutions, where the field components are eigenstates of a space-reflection symmetry, the first four and the sequence of even order charges are exactly conserved in the scattering process of the solitons. Such results are obtained through analytical and numerical methods, and employ adaptations of algebraic techniques used in integrable field theories. We perform extensive numerical simulations and consider the scattering of dark solitons for the cubic-quintic NLS model with potential \( V=\eta {I}^2-\frac{\in }{6}{I}^3 \) and the saturable type potential satisfying , with a deformation parameter ϵ ∈ and I = |ψ|2. The issue of the renormalization of the charges and anomalies, and their (quasi)conservation laws are properly addressed. The saturable NLS supports elastic scattering of two soliton solutions for a wide range of values of {η, ϵ, q}. Our results may find potential applications in several areas of non-linear science, such as the Bose-Einstein condensation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L.A. Ferreira and W.J. Zakrzewski, The concept of quasi-integrability: a concrete example, JHEP 05 (2011) 130 [arXiv:1011.2176] [INSPIRE].
L.A. Ferreira and W.J. Zakrzewski, Numerical and analytical tests of quasi-integrability in modified sine-Gordon models, JHEP 01 (2014) 058 [arXiv:1308.4412] [INSPIRE].
V.H. Aurichio and L.A. Ferreira, Quasi-Integrable Deformations of the Bullough-Dodd model, JHEP 03 (2015) 152 [arXiv:1501.01821] [INSPIRE].
L.A. Ferreira, G. Luchini and W.J. Zakrzewski, The concept of quasi-integrability for modified non-linear Schrödinger models, JHEP 09 (2012) 103 [arXiv:1206.5808] [INSPIRE].
J.P. Keener and D.W. McLaughlin, Solitons under perturbations, Phys. Rev. A 16 (1977) 777. [Erratum ibid. A 17 (1978) 1555].
J.P. Keener and D.W. Mclaughlin, A Green’s Function for a Linear Equation Associated with Solitons, J. Math. Phys. 18 (1977) 2008 [INSPIRE].
B.A. Malomed, Inelastic interactions of solitons in nearly integrable systems. I, Physica D 15 (1985) 374.
M.J. Ablowitz, S.D. Nixon, T.P. Horikis and D.J. Frantzeskakis, Perturbations of Dark Solitons, Proc. Roy. Soc. Lond. A 467 (2011) 2597 [arXiv:1008.3756] [INSPIRE].
Yu.S. Kivshar and B. Luther-Davies, Dark optical solitons: physics and applications, Phys. Rept. 298 (1998) 81.
X.-J.Chen, Z.-D. Chen and N.-N. Huang, A direct perturbation theory for dark solitons based on a complete set of the squared Jost solutions, J. Phys. A 31 (1998) 6929.
V.M. Lashkin, Perturbation theory for dark solitons: Inverse scattering transform approach and radiative effects, Phys. Rev. E 70 (2004) 066620.
S.-M. Ao and J.-R. Yan, A perturbation method for dark solitons based on a complete set of the squared Jost solutions, J. Phys. A 38 (2005) 2399.
J.-L. Yu, Ch.-N. Yang, H. Cai and N.-N. Huang, Direct perturbation theory for the dark soliton solution to the nonlinear Schrödinger equation with normal dispersion, Phys. Rev. E 75 (2007) 046604.
H. Blas and M. Zambrano, Spatial shifts of colliding dark solitons in deformed non-linear Schrödinger models, J. Phys. A 48 (2015) 275201.
G. Theocharis et al., Multiple atomic dark solitons in cigar-shaped Bose-Einstein condensates, Phys. Rev. A 81 (2010) 063604.
W. Bao, Numerical methods for the nonlinear Schrödinger equation with nonzero far-field conditions, Meth. Appl. Anal. 11 (2004) 367.
W. Bao, Q. Tang and Z. Xu, Numerical methods and comparison for computing dark and bright solitons in the nonlinear Schrödinger equation, J. Comput. Phys. 235 (2013) 423.
S. Cowan, R.H. Enns, S.S. Rangnekar and S.S. Sanghera, Quasi-soliton and other behaviour of the nonlinear cubic-quintic Schrödinger equation, Can. J. Phys. 64 (1986) 311
M. Crosta, A. Fratalocchi and S. Trillo, Bistability and instability of dark-antidark solitons in the cubic-quintic nonlinear Schrödinger equation, Phys. Rev. A 84 (2011) 063809
A.S.B. Sombra, Bistable pulse collisions of the cubic-quintic nonlinear Schrödinger equation, Opt. Commun. 94 (1992) 92.
W. Krolikowski and B. Luther-Davies, Dark optical solitons in saturable nonlinear media, Opt. Lett. 18 (1993) 188.
R.H. Enns, Bistable solitons and the Painlevé test, Phys. Rev. A 36 (1987) 5441.
D. Chiron, Travelling waves for the nonlinear Schrödinger equation with general nonlinearity in dimension one, Nonlinearity 25 (2012) 813.
F.G. Bass, V.V. Konotop and S.A. Puzenko, Dark solitary waves in a generalized version of the nonlinear Schrödinger equation, Phys. Rev. A 46 (1992) 4185.
A. de O. Assunção, H. Blas and M.J.B.F. da Silva, New derivation of soliton solutions to the AKNS 2 system via dressing transformation methods, J. Phys. A 45 (2012) 085205.
Y. Ohta, D.-S. Wang and J. Yang, General N-dark-dark solitons in the coupled nonlinear Schrödinger equations, Stud. Appl. Math. 127 (2011) 345.
A.M. Kamchatnov and M. Salerno, Dark soliton oscillations in Bose-Einstein condensates with multi-body interactions, J. Phys. B 42 (2009) 185303.
D.J. Frantzeskakis, Dark solitons in atomic Bose-Einstein condensates: from theory to experiments, J. Phys. A 43 (2010) 213001.
J. Hietarinta, Hirota’s bilinear method and partial integrability, in Partially Integrable Equations in Physics, R. Conte and N. Boccara eds., Les Houches, France 21-30 March 1989, NATO ASI Ser. C 310 (1990) 459.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1511.04748
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Blas, H., Zambrano, M. Quasi-integrability in the modified defocusing non-linear Schrödinger model and dark solitons. J. High Energ. Phys. 2016, 5 (2016). https://doi.org/10.1007/JHEP03(2016)005
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2016)005