We investigate the dynamics of dark optical solitons of the defocusing nonlinear Schrödinger (NLS) equation under the influence of small perturbations. Building on the integrable structure of the unperturbed NLS equation, we present two complementary analytical frameworks: a perturbed conservation-law approach and a linearization-based eigenfunction expansion rooted in inverse scattering theory. Both yield consistent evolution equations for the key soliton parameters—velocity, depth, center and phase—but the eigenfunction approach additionally provides an explicit expression for the firstorder correction, from which the structure of the shelf, a plateau-like radiation field that inevitably develops around the soliton due to the nonvanishing boundary conditions, can be extracted. It is found that the existence of the shelf is characterized by a single parameter α, which depends on the form of the perturbation. A central result of this work is an exact algebraic relation between the shelf parameter α and the renormalized soliton energy: the rate of energy change is directly and exclusively coupled to α, while all other macroscopic conserved quantities—momentum, Hamiltonian, and center of energy—are shown to decouple from the shelf at leading order. This energy–shelf relation provides a unified and physically transparent accounting of how perturbation energy is partitioned between the soliton core and the radiation it emits. Our analytical results—which are illustrated through several examples of physically relevant perturbations—are consistent with established results in limiting cases. To our knowledge, the energy–shelf relation and the systematic comparison of the two perturbative frameworks within a single, unified treatment are presented here for the first time.
- Open Access
- Article
Perturbed Dark Optical Solitons
- Nicholas J. Ossi 1,
- Barbara Prinari 1,2,
- Theodoros P. Horikis 2,
- Dimitrios J. Frantzeskakis 3,*
Author Information
Received: 07 May 2026 | Revised: 08 Jul 2026 | Accepted: 09 Jul 2026 | Published: 20 Jul 2026
Abstract
Keywords
dark solitons | defocusing nonlinear Schrödinger equation | perturbation theory | inverse scattering transform | radiation shelf
References
- 1.
Zakharov, V.E.; Shabat, A.B. Interaction between solitons in a stable medium. Sov. Phys. JETP. 1973, 37, 823–828.
- 2.
Tsuzuki, T.Nonlinearwaves in the Pitaevskii-Gross equation. J. LowTemp. Phys. 1971, 4, 441–457. https://doi.org/10.1007/BF00628744.
- 3.
Hasegawa, A.; Tappert, F. Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion. Appl. Phys. Lett. 1973, 23, 171–172. https://doi.org/10.1063/1.1654847.
- 4.
Emplit, E.; Hamaide, J.E.; Reynaud, E.; et al. Picosecond steps and dark pulses through nonlinear single-mode fibers. Opt. Commun. 1987, 62, 374–379. https://doi.org/10.1016/0030-4018(87)90003-4.
- 5.
Krokel, D.; Halas, N.J.; Giuliani, G.; et al. Dark-pulse propagation in optical fibers. Phys. Rev. Lett. 1988, 60, 29–32. https://doi.org/10.1103/PhysRevLett.60.29.
- 6.
Weiner, A.M.; Heritage, J.P.; Hawkins, R.J.; et al. Experimental observation of the fundamental dark soliton in optical fibers. Phys. Rev. Lett. 1988, 61, 2445–2448. https://doi.org/10.1103/PhysRevLett.61.2445.
- 7.
Hasegawa, A.; Kodama, Y. Solitons in Optical Communications; Clarendon Press: Oxford, UK, 1995. https://doi.org/10.1093/oso/9780198565079.001.0001.
- 8.
Kivshar, Yu.S.; Luther-Davies, B. Dark optical solitons: Physics and applications. Phys. Rep. 1998, 298, 81–197. https://doi.org/10.1016/S0370-1573(97)00073-2.
- 9.
Agrawal, G.P. Applications of Nonlinear Fiber Optics; Academic: San Diego, CA, USA, 2008. https://doi.org/10.1016/B978-0-12-374302-2.X5001-3.
- 10.
Kivshar, Yu.S.; Agrawal, G.P. Optical Solitons: From Fibers to Photonic Crystals; Academic Press: San Diego, CA, USA, 2003. https://doi.org/10.1016/B978-0-12-410590-4.X5000-1.
- 11.
Kivshar, Yu.S.; Malomed, B.A. Dynamics of solitons in nearly integrable systems. Rev. Mod. Phys. 1989, 61, 763. https://doi.org/10.1103/RevModPhys.61.763.
- 12.
Zhao,W.; Bourkoff, E. Propagation properties of dark solitons. Opt. Lett. 1989, 14, 703–705. https://doi.org/10.1364/OL.14.000703.
- 13.
Giannini, J.A.; Joseph, R.I. The propagation of bright and dark solitons in lossy optical fibers. IEEE J. Quantum Electron. 1990, 26, 2109–2114. https://doi.org/10.1109/3.64346.
- 14.
Lisak, M.; Anderson, D.; Malomed, B.A. Dissipative damping of dark solitons in optical fibers. Opt. Lett. 1991, 16, 1936–1937. https://doi.org/10.1364/OL.16.001936.
- 15.
Kivshar, Yu.S.; Yang, X.P. Perturbation-induced dynamics of dark solitons. Phys. Rev. E 1994, 49, 1657–1670. https://doi.org/10.1103/PhysRevE.49.1657.
- 16.
Frantzeskakis, D.J. Dark solitons in atomic Bose-Einstein condensates: from theory to experiments. J. Phys. A Math. Theor. 2010, 43, 213001. https://doi.org/10.1088/1751-8113/43/21/213001.
- 17.
Kevrekidis, P.G.; Frantzeskakis, D.J.; Carretero-Gonzalez, R. The Defocusing Nonlinear Schrodinger Equation: From Dark Solitons to Vortices and Vortex Rings; SIAM: Philadelphia, PA, USA, 2015. https://doi.org/10.1137/1.9781611973945.
- 18.
Ablowitz, M.J.; Segur, H. Solitons and the Inverse Scattering Transform; SIAM: Philadelphia, PA, USA, 1981. https://doi.org/10.1137/1.9781611970883.
- 19.
Konotop, V.V.; Vekslerchik, V.E. Direct perturbation-theory for dark solitons. Phys. Rev. E 1994, 49, 2397–2407. https://doi.org/10.1103/PhysRevE.49.2397.
- 20.
Kaup, D.J. A perturbation expansion for the Zakharov-Shabat inverse scattering transform. SIAM J. Appl. Math. 1976, 31, 121–133. https://doi.org/10.1137/0131013.
- 21.
Chen, X.-J.; Chen, Z.-D.; Huang, N.-N. A direct perturbation theory for dark solitons based on a complete set of the squared Jost solutions. J. Phys. A Math. Gen. 1998, 31, 6929–6947. https://doi.org/10.1088/0305-4470/31/33/005.
- 22.
Chen, X.-J.; Chen, Z.-D. Dark optical solitons on influence of the self-steepening term. Chin. Phys. Lett. 1998, 15, 504–506.
- 23.
Lashkin, V.M. Perturbation theory for dark solitons: Inverse scattering transform approach and radiative effects. Phys. Rev. E 2004, 70, 066620. https://doi.org/10.1103/PhysRevE.70.066620.
- 24.
Ao, S.-M.; Yan, J.-R. A perturbation method for dark solitons based on a complete set of the squared Jost solutions. J. Phys. A Math. Gen. 2005, 38, 2399–2413. https://doi.org/10.1088/0305-4470/38/11/006.
- 25.
Kodama, Y.; Ablowitz, M.J. Perturbations of solitons and solitary waves. Stud. Appl. Math. 1981, 64, 225–245. https://doi.org/10.1002/sapm1981643225.
- 26.
Burtsev, S.; Camassa, R. Nonadiabatic dynamics of dark solitons. J. Opt. Soc. Am. B 1997, 14, 1782–1787. http://doi.org/10.1364/JOSAB.14.001782.
- 27.
Ablowitz, M.J.; Nixon, S.D.; Horikis, T.P.; et al. Perturbations of dark solitons. Proc. Roy. Soc. A 2011, 467, 2597–2621. https://doi.org/10.1098/rspa.2010.0663.
- 28.
Horikis, T.P.; Frantzeskakis, D.J. Dark solitons in the presence of higher-order effects. Opt. Lett. 2013, 38, 5098–5101. https://doi.org/10.1364/OL.38.005098.
- 29.
Horikis, T.P. Dark soliton dynamics under the complex Ginzburg-Landau equation. Chaos Solitions Fract. 2015, 77, 94–100. https://doi.org/10.1016/j.chaos.2015.04.019.
- 30.
Ossi, N.J.; Prinari, B.; Yang, J. Integrable perturbation theory for dark solitons of the defocusing nonlinear Schrodinger equation. J. Nonlinear Waves 2026, 2, e10. https://doi.org/10.1017/jnw.2026.10035.
- 31.
Prinari, B. Inverse Scattering Transform for nonlinear Schrodinger system on a nontrivial background: A survey of classical results, new developments and future directions. J. Nonlin. Math. Phys. 2023, 30, 317–383. https://doi.org/10.1007/s44198-023-00120-w.
- 32.
Chernyavskiy, A.; Frantzeskakis, D.J.; Horikis, T.P.; et al. Dark-bright soliton perturbation theory for the Manakov system. Proc. Roy. Soc. A 2025, 481, 20240082. https://doi.org/10.1098/rspa.2024.0082.
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