A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers

  • Boris A. MalomedEmail author
Reference work entry


The chapter provides a survey of (chiefly, theoretical) results obtained for self-trapped modes (solitons) in various models of one-dimensional optical waveguides based on a pair of parallel guiding cores, which combine the linear inter-core coupling with the intrinsic cubic (Kerr) nonlinearity, anomalous group-velocity dispersion, and, possibly, intrinsic loss and gain in each core. The survey is focused on three main topics: spontaneous breaking of the inter-core symmetry and the formation of asymmetric temporal solitons in dual-core fibers; stabilization of dissipative temporal solitons (essentially, in the model of a fiber laser) by a lossy core parallel-coupled to the main one, which carries the linear gain; and stability conditions for \(\mathcal{P}\mathcal{T}\) (parity-time)-symmetric solitons in the dual-core nonlinear dispersive coupler with mutually balanced linear gain and loss applied to the two cores.



I thank Professor Gang-Ding Peng for his invitation to join the production of this volume and to write the present chapter.


  1. F.K. Abdullaev, R.M. Abrarov, S.A. Darmanyan, Dynamics of solitons in coupled optical fibers. Opt. Lett. 14, 131–133 (1989)CrossRefGoogle Scholar
  2. A.B. Aceves, Optical gap solitons: past, present, and future; theory and experiments. Chaos 10, 584–589 (2000)CrossRefGoogle Scholar
  3. A.B. Aceves, S. Wabnitz, Self-induced transparency solitons in nonlinear refractive periodic media. Phys. Lett. A 141, 37–42 (1989)CrossRefGoogle Scholar
  4. A.B. Aceves, C. De Angelis, A.M. Rubenchik, S.K. Turitsyn, Multidimensional solitons in fiber arrays. Opt. Lett. 19, 329–331 (1994a)CrossRefGoogle Scholar
  5. A.B. Aceves, C. De Angelis, G.G. Luther, A.M. Rubenchik, Multidimensional solitons in fiber arrays. Opt. Lett. 19, 1186–1188 (1994b)CrossRefGoogle Scholar
  6. A.B. Aceves, G.G. Luther, C. De Angelis, A.M. Rubenchik, S.K. Turitsyn, Energy localization in nonlinear fiber arrays: collapse-effect compressor. Phys. Rev. Lett. 75, 73–76 (1995)CrossRefGoogle Scholar
  7. V.V. Afanasjev, B.A. Malomed, P.L. Chu, Dark soliton generation in a fused coupler. Opt. Commun. 137, 229–232 (1997)CrossRefGoogle Scholar
  8. G.P. Agrawal, Nonlinear Fiber Optics, 4th edn. (Academic, San Diego, 2007)Google Scholar
  9. N. Akhmediev, A. Ankiewicz, Novel soliton states and bifurcation phenomena in nonlinear fiber couplers. Phys. Rev. Lett. 70, 2395–2398 (1993a)CrossRefGoogle Scholar
  10. N. Akhmediev, A. Ankiewicz, Spatial soliton X-junctions and couplers. Opt. Commun. 100, 186–192 (1993b)CrossRefGoogle Scholar
  11. L. Albuch, B.A. Malomed, Transitions between symmetric and asymmetric solitons in dual-core systems with cubic-quintic nonlinearity. Math. Comput. Simul. 74, 312–322 (2007)CrossRefGoogle Scholar
  12. N.V. Alexeeva, I.V. Barashenkov, A.A. Sukhorukov, Y.S. Kivshar, Optical solitons in \(\mathcal{P}\mathcal{T}\)-symmetric nonlinear couplers with gain and loss. Phys. Rev. A 85, 063837 (2012)Google Scholar
  13. D. Anderson, Variational approach to nonlinear pulse propagation in optical fibers. Phys. Rev. A 27, 3135–3145 (1983)CrossRefGoogle Scholar
  14. D. Anderson, M. Lisak, T. Reichel, Asymptotic propagation properties of pulses in a soliton-based optical-fiber communication system. J. Opt. Soc. Am. B 5, 207–210 (1988)CrossRefGoogle Scholar
  15. A. Ankiewicz, N. Akhmediev, G.D. Peng, P.L. Chu, Limitations of the variational approach in soliton propagation in nonlinear couplers. Opt. Commun. 103, 410 (1993)CrossRefGoogle Scholar
  16. I.S. Aranson, L. Kramer, The world of the complex Ginzburg-Landau equation. Rev. Mod. Phys. 74, 99–143 (2002)CrossRefGoogle Scholar
  17. G. Arjunan, B.A. Malomed, M. Arumugam, U. Ambikapathy, Modulational instability in linearly coupled asymmetric dual-core fibers. Appl. Sci. 7, 645 (2017)CrossRefGoogle Scholar
  18. J. Atai, B.A. Malomed, Stability and interactions of solitons in two-component systems. Phys. Rev. E 54, 4371–4374 (1996)CrossRefGoogle Scholar
  19. J. Atai, B.A. Malomed, Bound states of solitary pulses in linearly coupled Ginzburg-Landau equations. Phys. Lett. A 244, 551–556 (1998a)CrossRefGoogle Scholar
  20. J. Atai, B.A. Malomed, Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations. Phys. Lett. A 246, 412–422 (1998b)CrossRefGoogle Scholar
  21. J. Atai, B.A. Malomed, Bragg-grating solitons in a semilinear dual-core system. Phys. Rev. E 62, 8713–8718 (2000)CrossRefGoogle Scholar
  22. I.V. Barashenkov, D.E. Pelinovsky, E.V. Zemlyanaya, Vibrations and oscillatory instabilities of gap solitons. Phys. Rev. Lett. 80, 5117 (1998)CrossRefGoogle Scholar
  23. C.M. Bender, Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947 (2007)CrossRefGoogle Scholar
  24. C.M. Bender, S. Boettcher, Real spectra in non-Hermitian Hamiltonians having \(\mathcal{P}\mathcal{T}\) symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)CrossRefGoogle Scholar
  25. M.V. Berry, Optical lattices with \(\mathcal{P}\mathcal{T}\) symmetry are not transparent. J. Phys. A 41, 244007 (2008)Google Scholar
  26. R. Blit, B.A. Malomed, Propagation and collisions of semidiscrete solitons in arrayed and stacked waveguides. Phys. Rev. A 86, 043841 (2012)CrossRefGoogle Scholar
  27. A. Boskovic, S.V. Chernikov, J.R. Taylor, Spectral filtering effect of fused fiber couplers in femtosecond fiber soliton lasers. J. Mod. Opt. 42, 1959–1963 (1995)CrossRefGoogle Scholar
  28. G. Burlak, B.A. Malomed, Stability boundary and collisions of two-dimensional solitons in \(\mathcal{P}\mathcal{T}\)-symmetric couplers with the cubic-quintic nonlinearity. Phys. Rev. E 88, 062904 (2013)Google Scholar
  29. G. Burlak, S. Garcia-Paredes, B.A. Malomed, \(\mathcal{P}\mathcal{T}\)-symmetric couplers with competing cubic-quintic nonlinearities. Chaos 26, 113103 (2016)CrossRefGoogle Scholar
  30. A.R. Champneys, B.A. Malomed, J. Yang, D.J. Kaup, “Embedded solitons”: solitary waves in resonance with the linear spectrum. Phys. D 152–153, 340–354 (2001)CrossRefGoogle Scholar
  31. D. Chevriaux, R. Khomeriki, J. Leon, Bistable transmitting nonlinear directional couplers. Mod. Phys. Lett. B 20, 515–532 (2006)CrossRefGoogle Scholar
  32. K.S. Chiang, Intermodal dispersion in two-core optical fibers. Opt. Lett. 20, 997–999 (1995)CrossRefGoogle Scholar
  33. D.N. Christodoulides, R.I. Joseph, Slow Bragg solitons in nonlinear periodic structures. Phys. Rev. Lett. 62, 1746–1749 (1989)CrossRefGoogle Scholar
  34. P.L. Chu, B.A. Malomed, G.D. Peng, Soliton switching and propagation in nonlinear fiber couplers: analytical results. J. Opt. Soc. Am. B 10, 1379–1385 (1993)CrossRefGoogle Scholar
  35. P.L. Chu, G.D. Peng, B.A. Malomed, H. Hatami-Hansa, I.M. Skinner, Time domain soliton filter based on a semidissipative dual-core coupler. Opt. Lett. 20, 1092–1094 (1995a)CrossRefGoogle Scholar
  36. P.L. Chu, Y.S. Kivshar, B.A. Malomed, G.D. Peng, M.L. Quiroga-Teixeiro, Soliton controlling, switching, and splitting in fused nonlinear couplers. J. Opt. Soc. Am. B 12, 898–903 (1995b)CrossRefGoogle Scholar
  37. P.L. Chu, B.A. Malomed, G.D. Peng, Passage of a pulse through a nonlinear amplifier. Opt. Commun. 140, 289–295 (1997)CrossRefGoogle Scholar
  38. G. Cohen, Soliton interaction and stability in nonlinear directional fiber couplers. Phys. Rev. E 52, 5565–5573 (1995)CrossRefGoogle Scholar
  39. S. Cowan, R.H. Enns, S.S. Rangnekar, S.S. Sanghera, Quasi-soliton and other behaviour of the nonlinear cubic-quintic Schrödinger equation. Can. J. Phys. 64, 311–315 (1986)CrossRefGoogle Scholar
  40. A. De Rossi, C. Conti, S. Trillo, Stability, multistability, and wobbling of optical gap solitons. Phys. Rev. Lett. 81, 85–88 (1998)CrossRefGoogle Scholar
  41. C.M. de Sterke, J.E. Sipe, Gap solitons. Prog. Opt. 33, 203–260 (1994)CrossRefGoogle Scholar
  42. M.J.F. Digonnet, H.J. Shaw, Analysis of a tunable single-mode optical fiber coupler. IEEE J. Quantum Electron 18, 746–754 (1982)CrossRefGoogle Scholar
  43. S.L. Doty, J.W. Haus, Y. Oh, R.L. Fork, Soliton interactions on dual-core fibers. Phys. Rev. E 51, 709–717 (1995)CrossRefGoogle Scholar
  44. R. Driben, B.A. Malomed, Stability of solitons in parity–time-symmetric couplers. Opt. Lett. 36, 4323–4325 (2011a)CrossRefGoogle Scholar
  45. R. Driben, B.A. Malomed, Stabilization of solitons in \(\mathcal{P}\mathcal{T}\) models with supersymmetry by periodic management. EPL 96, 51001 (2011b)CrossRefGoogle Scholar
  46. N. Dror, B.A. Malomed, Symmetric and asymmetric solitons and vortices in linearly coupled two-dimensional waveguides with the cubic-quintic nonlinearity. Phys. D 240, 526–541 (2011)CrossRefGoogle Scholar
  47. N. Efremidis, K. Hizanidis, B.A. Malomed, H.E. Nistazakis, D.J. Frantzeskakis, Stable transmission of solitons in the region of normal dispersion. J. Opt. Soc. Am. B 17, 952–958 (2000a)CrossRefGoogle Scholar
  48. N. Efremidis, K. Hizanidis, H.E. Nistazakis, D.J. Frantzeskakis, B.A. Malomed, Stabilization of dark solitons in the cubic Ginzburg-Landau equation. Phys. Rev. E 62, 7410–7414 (2000b)CrossRefGoogle Scholar
  49. F. Eilenberger, K. Prater, S. Minardi, R. Geiss, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, T. Pertsch, Observation of discrete, vortex light bullets. Phys. Rev. X 3, 041031 (2013)Google Scholar
  50. G.A. El, R.H.G. Grimshaw, N.F. Smyth, Unsteady undular bores in fully nonlinear shallow-water theory. Phys. Fluids 18, 027104 (2006)CrossRefGoogle Scholar
  51. R. El-Ganainy, K.G. Makris, D.N. Christodoulides, Z.H. Musslimani, Theory of coupled optical \(\mathcal{P}\mathcal{T}\)-symmetric structures. Opt. Lett. 32, 2632–2634 (2007)CrossRefGoogle Scholar
  52. A. Espinosa-Ceron, B.A. Malomed, J. Fujioka, R.F. Rodriguez, Symmetry breaking in linearly coupled KdV systems. Chaos 22, 033145 (2012)CrossRefGoogle Scholar
  53. W.J. Firth, P.V. Paulau, Soliton lasers stabilized by coupling to a resonant linear system. Eur. Phys. J. D 59, 13–21 (2010)CrossRefGoogle Scholar
  54. S.R. Friberg, Y. Silberberg, M.K. Oliver, M.J. Andrejco, M.A. Saifi, P.W. Smith, Ultrafast all-optical switching in dual-core fiber nonlinear coupler. Appl. Phys. Lett. 51, 1135–1137 (1987)CrossRefGoogle Scholar
  55. S.R. Friberg, A.M. Weiner, Y. Silberberg, B.G. Sfez, P.S. Smith, Femtosecond switching in dual-core-fiber nonlinear coupler. Opt. Lett. 13, 904–906 (1988)CrossRefGoogle Scholar
  56. R. Ganapathy, B.A. Malomed, K. Porsezian, Modulational instability and generation of pulse trains in asymmetric dual-core nonlinear optical fibers. Phys. Lett. A 354, 366–372 (2006)CrossRefGoogle Scholar
  57. J.A. Gear, R. Grimshaw, Weak and strong-interactions between internal solitary waves. Stud. Appl. Math. 70, 235–258 (1984)CrossRefGoogle Scholar
  58. P. Grelu, N. Akhmediev, Dissipative solitons for mode-locked lasers. Nat. Photonics 6, 84–92 (2012)CrossRefGoogle Scholar
  59. A. Gubeskys, B.A. Malomed, Symmetric and asymmetric solitons in linearly coupled Bose-Einstein condensates trapped in optical lattices. Phys. Rev. A 75, 063602 (2007)CrossRefGoogle Scholar
  60. A. Guo, G.J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G.A. Siviloglou, D.N. Christodoulides, Observation of \(\mathcal{P}\mathcal{T}\)-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902 (2009)Google Scholar
  61. Lj. Hadžievski, G. Gligorić, A. Maluckov, B.A. Malomed, Interface solitons in one-dimensional locally coupled lattice systems. Phys. Rev. A 82, 033806 (2010)Google Scholar
  62. V. Hakim, P. Jakobsen, Y. Pomeau, Fronts vs. solitary waves in nonequilibrium systems. Europhys. Lett. 11, 19–24 (1990)CrossRefGoogle Scholar
  63. A. Harel, B.A. Malomed, Interactions of spatial solitons with fused couplers. Phys. Rev. A 89, 043809 (2014)CrossRefGoogle Scholar
  64. H. Hatami-Hanza, P.L. Chu, B.A. Malomed, G.D. Peng, Soliton compression and splitting in double-core nonlinear optical fibers. Opt. Commun. 134, 59–65 (1997)CrossRefGoogle Scholar
  65. D.R. Heatley, E.M. Wright, G.I. Stegeman, Soliton coupler. Appl. Phys. Lett. 53, 172–174 (1988)CrossRefGoogle Scholar
  66. G. Herring, P.G. Kevrekidis, B.A. Malomed, R. Carretero-González, D.J. Frantzeskakis, Symmetry breaking in linearly coupled dynamical lattices. Phys. Rev. E 76, 066606 (2007)CrossRefGoogle Scholar
  67. M. Hochberg, T. Baehr-Jones, C. Walker, A. Scherer, Integrated plasmon and dielectric waveguides. Opt. Exp. 12, 5481–5486 (2004)CrossRefGoogle Scholar
  68. L.M. Hocking, K. Stewartson, On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance. Proc. R. Soc. Lond. A 326, 289–313 (1972)CrossRefGoogle Scholar
  69. W.P. Huang, Coupled-mode theory for optical waveguides: an overview. J. Opt. Soc. Am. A 11, 963–983 (1994)CrossRefGoogle Scholar
  70. G. Iooss, D.D. Joseph, Elementary Stability and Bifurcation Theory (Springer, Berlin, 1980)CrossRefGoogle Scholar
  71. S.M. Jensen, The nonlinear coherent coupler. IEEE J. Quantum Electron 18, 1580–1583 (1982); A.A. Maier, Optical transistors and bistable devices utilizing nonlinear transmission of light in systems with unidirectional coupled waves. Sov. J. Quantum Electron 12, 1490–1494 (1982)Google Scholar
  72. Y.V. Kartashov, B.A. Malomed, V.V. Konotop, V.E. Lobanov, L. Torner, Stabilization of solitons in bulk Kerr media by dispersive coupling. Opt. Lett. 40, 1045–1048 (2015)CrossRefGoogle Scholar
  73. D.J. Kaup, B.A. Malomed, Gap solitons in asymmetric dual-core nonlinear optical fibers. J. Opt. Soc. Am. B 15, 2838–2846 (1998)CrossRefGoogle Scholar
  74. D.J. Kaup, T.I. Lakoba, B.A. Malomed, Asymmetric solitons in mismatched dual-core optical fibers. J. Opt. Soc. Am. B 14, 1199–1206 (1997)CrossRefGoogle Scholar
  75. Y.S. Kivshar, B.A. Malomed, Dynamics of fluxons in a system of coupled Josephson junctions. Phys. Rev. B 37, 9325–9330 (1988)CrossRefGoogle Scholar
  76. Y.S. Kivshar, B.A. Malomed, Dynamics of solitons in nearly integrable systems. Rev. Mod. Phys. 61, 763–915 (1989a)CrossRefGoogle Scholar
  77. Y.S. Kivshar, B.A. Malomed, Interaction of solitons in tunnel-coupled optical fibers. Opt. Lett. 14, 1365–1367 (1989b)CrossRefGoogle Scholar
  78. S. Klaiman, U. Günther, N. Moiseyev, Visualization of branch points in \(\mathcal{P}\mathcal{T}\)-symmetric waveguides. Phys. Rev. Lett. 101, 080402 (2008)Google Scholar
  79. V.V. Konotop, J. Yang, D.A. Zezyulin, Nonlinear waves in \(\mathcal{P}\mathcal{T}\)-symmetric systems. Rev. Mod. Phys. 88, 035002 (2016)Google Scholar
  80. W. Królikowski, Y.S. Kivshar, Soliton-based optical switching in waveguide arrays. J. Opt. Soc. Am. B 13, 876–887 (1996)CrossRefGoogle Scholar
  81. T.I. Lakoba, D.J. Kaup, Stability of solitons in nonlinear fiber couplers with two orthogonal polarizations. Phys. Rev. E 56, 4791–4802 (1997)CrossRefGoogle Scholar
  82. T.I. Lakoba, D.J. Kaup, B.A. Malomed, Solitons in nonlinear fiber couplers with two orthogonal polarizations. Phys. Rev. E 55, 6107–6120 (1997)CrossRefGoogle Scholar
  83. L.D. Landau, E.M. Lifshitz, Mechanics (Nauka Publishers, Moscow, 1988)Google Scholar
  84. L.D. Landau, E.M. Lifshitz, Quantum Mechanics (Nauka Publishers, Moscow, 1989)Google Scholar
  85. F. Lederer, G.I. Stegeman, D.N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, Discrete solitons in optics. Phys. Rep. 463, 1–126 (2008)CrossRefGoogle Scholar
  86. C. Li, G. Xu, L. Ma, N. Dou, H. Gu, An erbium-doped fibre nonlinear coupler with coupling ratios controlled by pump power. J. Opt. A Pure Appl. Opt. 7, 540–543 (2005)CrossRefGoogle Scholar
  87. J.H. Li, K.S. Chiang, K.W. Chow, Modulation instabilities in two-core optical fibers. J. Opt. Soc. Am. B 28, 1693–1701 (2011)CrossRefGoogle Scholar
  88. Y. Li, W. Pang, S. Fu, B.A. Malomed, Two-component solitons under a spatially modulated linear coupling: inverted photonic crystals and fused couplers. Phys. Rev. A 85, 053821 (2012)CrossRefGoogle Scholar
  89. P. Li, L. Li, B.A. Malomed, Multisoliton Newton’s cradles and supersolitons in regular and parity-time-symmetric nonlinear couplers. Phys. Rev. E 89, 062926 (2014)CrossRefGoogle Scholar
  90. S. Longhi, Bloch oscillations in complex crystals with \(\mathcal{P}\mathcal{T}\) symmetry. Phys. Rev. Lett. 103, 123601 (2009)Google Scholar
  91. S.Y. Lou, B. Tong, H.C. Hu, X.Y. Tang, Coupled KdV equations derived from two-layer fluids. J. Phys. A Math. Gen. 39, 513–527 (2006)CrossRefGoogle Scholar
  92. W.N. MacPherson, J.D.C. Jones, B.J. Mangan, J.C. Knight, P.S.J. Russell, Two-core photonic crystal fibre for Doppler difference velocimetry. Opt. Commun. 223, 375–380 (2003)CrossRefGoogle Scholar
  93. A.I. Maimistov, Propagation of a light pulse in nonlinear tunnel-coupled optical waveguides. Kvantovaya Elektron (Moscow) 18, 758–761 (1991) [Sov. J. Quantum Electron 21, 687–690 (1991)]Google Scholar
  94. W.C.K. Mak, B.A. Malomed, P.L. Chu, Soliton coupling in waveguide with quadratic nonlinearity. Phys. Rev. E 55, 6134–6140 (1997)CrossRefGoogle Scholar
  95. W.C.K. Mak, B.A. Malomed, P.L. Chu, Solitary waves in coupled nonlinear waveguides with Bragg gratings. J. Opt. Soc. Am. B 15, 1685–1692 (1998)CrossRefGoogle Scholar
  96. Y. Makhlin, G. Schön, A. Shnirman, Quantum-state engineering with Josephson-junction devices. Rev. Mod. Phys. 73, 357–400 (2001)CrossRefGoogle Scholar
  97. K.G. Makris, R. El-Ganainy, D.N. Christodoulides, Z.H. Musslimani, Beam dynamics in \(\mathcal{P}\mathcal{T}\) symmetric optical lattices. Phys. Rev. Lett. 100, 103904 (2008)Google Scholar
  98. K.G. Makris, R. El-Ganainy, D.N. Christodoulides, Z.H. Musslimani, \(\mathcal{P}\mathcal{T}\)-symmetric periodic optical potentials. Int. J. Theor. Phys. 50, 1019–1041 (2011)Google Scholar
  99. B.A. Malomed, Leapfrogging solitons in a system of coupled Korteweg – de Vries equations. Wave Motion 9, 401 (1987a)CrossRefGoogle Scholar
  100. B.A. Malomed, Evolution of nonsoliton and “quasiclassical” wavetrains in nonlinear Schrödinger and Korteweg – de Vries equations with dissipative perturbations. Phys. D 29, 155–172 (1987b)CrossRefGoogle Scholar
  101. B.A. Malomed, Optical domain walls. Phys. Rev. E 50, 1565–1571 (1994)CrossRefGoogle Scholar
  102. B.A. Malomed, Variational methods in fiber optics and related fields, in Progress in Optics, vol. 43, ed. by E. Wolf (North Holland, Amsterdam, 2002), pp. 71–193Google Scholar
  103. B.A. Malomed, Complex Ginzburg-Landau equation, in Encyclopedia of Nonlinear Science, ed. by A. Scott (Routledge, New York, 2005), pp. 157–160Google Scholar
  104. B.A. Malomed, Soliton Management in Periodic Systems (Springer, New York, 2006)Google Scholar
  105. B.A. Malomed, Solitary pulses in linearly coupled Ginzburg-Landau equations. Chaos 17, 037117 (2007)CrossRefGoogle Scholar
  106. B.A. Malomed, A.A. Nepomnyashchy, Kinks and solitons in the generalized Ginzburg-Landau equation. Phys. Rev. A 42, 6009–6014 (1990)CrossRefGoogle Scholar
  107. B.A. Malomed, R.S. Tasgal, Vibration modes of a gap soliton in a nonlinear optical medium. Phys. Rev. E 49, 5787–5796 (1994)CrossRefGoogle Scholar
  108. B.A. Malomed, H.G. Winful, Stable solitons in two-component active systems. Phys. Rev. E 53, 5365–5368 (1996)CrossRefGoogle Scholar
  109. B.A. Malomed, G.D. Peng, P.L. Chu, A nonlinear optical amplifier based on a dual-core fiber. Opt. Lett. 21, 330–332 (1996a)CrossRefGoogle Scholar
  110. B.A. Malomed, I.M. Skinner, P.L. Chu, G.D. Peng, Symmetric and asymmetric solitons in twin-core nonlinear optical fibers. Phys. Rev. E 53, 4084–4091 (1996b)CrossRefGoogle Scholar
  111. B. Mandal, A.R. Chowdhury, Solitary optical pulse propagation in fused fibre coupler – effect of Raman scattering and switching. Chaos, Solitons Fractals 24, 557–565 (2005)CrossRefGoogle Scholar
  112. P. Marcq, H. Chaté, R. Conte, Exact solutions of the one-dimensional quintic complex Ginzburg-Landau equation. Phys. D 73, 305–317 (1994)CrossRefGoogle Scholar
  113. A. Marini, D.V. Skryabin, B.A. Malomed, Stable spatial plasmon solitons in a dielectric-metal-dielectric geometry with gain and loss. Opt. Exp. 19, 6616–6622 (2011)CrossRefGoogle Scholar
  114. E. Marti-Panameno, L.C. Gomez-Pavon, A. Luis-Ramos, M.M. Mendez-Otero, M.D.I. Castillo, Self-mode-locking action in a dual-core ring fiber laser. Opt. Commun. 194, 409–414 (2001)CrossRefGoogle Scholar
  115. M. Matsumoto, S. Katayama, A. Hasegawa, Optical switching in nonlinear waveguide arrays with a longitudinally decreasing coupling coefficient. Opt. Lett. 20, 1758–1760 (1995)CrossRefGoogle Scholar
  116. M. Matuszewski, B.A. Malomed, M. Trippenbach, Spontaneous symmetry breaking of solitons trapped in a double-channel potential. Phys. Rev. A 75, 063621 (2007)CrossRefGoogle Scholar
  117. S. Minardi, F. Eilenberger, Y.V. Kartashov, A. Szameit, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, L. Torner, F. Lederer, A. Tünnermann, T. Pertsch, Three-dimensional light bullets in arrays of waveguides. Phys. Rev. Lett. 105, 263901 (2010)CrossRefGoogle Scholar
  118. M.B. Mineev, G.S. Mkrtchyan, V.V. Shmidt, On some effects in a system of 2 interacting Josephson junctions. J. Low Temp. Phys. 45, 497–505 (1981)CrossRefGoogle Scholar
  119. Z.H. Musslimani, K.G. Makris, R. El-Ganainy, D.N. Christodoulides, Optical solitons in \(\mathcal{P}\mathcal{T}\) periodic potentials. Phys. Rev. Lett. 100, 030402 (2008)Google Scholar
  120. H.E. Nistazakis, D.J. Frantzeskakis, J. Atai, B.A. Malomed, N. Efremidis, K. Hizanidis, Multi-channel pulse dynamics in a stabilized Ginzburg-Landau system. Phys. Rev. E 65, 036605 (2002)CrossRefGoogle Scholar
  121. K. Nozaki, N. Bekki, Exact solutions of the generalized Ginzburg-Landau equation. J. Phys. Soc. Jpn. 53, 1581–1582 (1984)CrossRefGoogle Scholar
  122. K. Ogusu, J. Yamasaki, S. Maeda, M. Kitao, M. Minakata, Linear and nonlinear optical properties of Ag-As-Se chalcogenide glasses for all-optical switching. Opt. Lett. 29, 265–267 (2004)CrossRefGoogle Scholar
  123. C. Paré, M. Florjańczyk, Approximate model of soliton dynamics in all-optical fibers. Phys. Rev. A 41, 6287–6295 (1990)CrossRefGoogle Scholar
  124. P.V. Paulau, D. Gomila, P. Colet, N.A. Loiko, N.N. Rosanov, T. Ackemann, W.J. Firth, Vortex solitons in lasers with feedback. Opt. Exp. 18, 8859–8866 (2010)CrossRefGoogle Scholar
  125. P.V. Paulau, D. Gomila, P. Colet, B.A. Malomed, W.J. Firth, From one- to two-dimensional solitons in the Ginzburg-Landau model of lasers with frequency-selective feedback. Phys. Rev. E 84, 036213 (2011)CrossRefGoogle Scholar
  126. G.D. Peng, P.L. Chu, A. Ankiewicz, Soliton propagation in saturable nonlinear fiber couplers – variational and numerical results. Int. J. Nonlin. Opt. Phys. 3, 69–87 (1994)CrossRefGoogle Scholar
  127. G.D. Peng, B.A. Malomed, P.L. Chu, Soliton collisions in a model of a dual-core nonlinear optical fiber. Phys. Scr. 58, 149–158 (1998)CrossRefGoogle Scholar
  128. N.R. Pereira, L. Stenflo, Nonlinear Schrödinger equation including growth and damping. Phys. Fluids 20, 1733–1734 (1977)CrossRefGoogle Scholar
  129. C.J. Pethick, H. Smith, Bose-Einstein Condensation in Dilute Gases, 2nd edn. (Cambridge University Press, Cambridge, 2008)CrossRefGoogle Scholar
  130. J. Petráček, Nonlinear directional coupling between plasmonic slot waveguides. Appl. Phys. B 112, 593–598 (2013)CrossRefGoogle Scholar
  131. V.I. Petviashvili, A.M. Sergeev, Spiral solitons in active media with excitation thresholds. Dokl. AN SSSR 276, 1380–1384 (1984) [Sov. Phys. Doklady 29, 493 (1984)]Google Scholar
  132. K.I. Pushkarov, D.I. Pushkarov, I.V. Tomov, Self-action of light beans in nonlinear media: soliton solutions. Opt. Quant. Electr. 11, 471–478 (1979)CrossRefGoogle Scholar
  133. V. Rastogi, K.S. Chiang, N.N. Akhmediev, Soliton states in a nonlinear directional coupler with intermodal dispersion. Phys. Lett. A 301, 27–34 (2002)CrossRefGoogle Scholar
  134. D.J. Richardson, J. Nilsson, W.A. Clarkson, High power fiber lasers: current status and future perspectives. J. Opt. Soc. Am. B 27, B63–B92 (2010)CrossRefGoogle Scholar
  135. M. Romagnoli, S. Trillo, S. Wabnitz, Soliton switching in nonlinear couplers. Opt. Quantum Electron 24, S1237–S1267 (1992)CrossRefGoogle Scholar
  136. A. Ruschhaupt, F. Delgado, J.G. Muga, Physical realization of \(\mathcal{P}\mathcal{T}\)-symmetric potential scattering in a planar slab waveguide. J. Phys. A Math. Gen. 38, L171 (2005)Google Scholar
  137. C.E. Ruter, K.G. Makris, R. El-Ganainy, D.N. Christodoulides, M. Segev, D. Kip, Observation of parity-time symmetry in optics. Nat. Phys. 6, 192–195 (2010)CrossRefGoogle Scholar
  138. J.P. Sabini, N. Finalyson, G.I. Stegeman, All-optical switching in nonlinear X junctions. Appl. Phys. Lett. 55, 1176–1178 (1989)CrossRefGoogle Scholar
  139. K. Saitoh, Y. Sato, M. Koshiba, Coupling characteristics of dual-core photonic crystal fiber couplers. Opt. Exp. 11, 3188–3195 (2003)CrossRefGoogle Scholar
  140. H. Sakaguchi, Hole solutions in the complex Ginzburg-Landau equation near a subcritical bifurcation. Progr. Theor. Phys. 86, 7–12 (1991)CrossRefGoogle Scholar
  141. H. Sakaguchi, B.A. Malomed, Breathing and randomly walking pulses in a semilinear Ginzburg-Landau system. Phys. D 147, 273–282 (2000)CrossRefGoogle Scholar
  142. H. Sakaguchi, B.A. Malomed, Symmetry breaking of solitons in two-component Gross-Pitaevskii equations. Phys. Rev. E 83, 036608 (2011)CrossRefGoogle Scholar
  143. H. Sakaguchi, B.A. Malomed, One- and two-dimensional solitons in \(\mathcal{P}\mathcal{T}\)-symmetric systems emulating spin–orbit coupling. New J. Phys. 18, 105005 (2016)CrossRefGoogle Scholar
  144. L. Salasnich, B.A. Malomed, F. Toigo, Competition between the symmetry breaking and onset of collapse in weakly coupled atomic condensates. Phys. Rev. A 81, 045603 (2010)CrossRefGoogle Scholar
  145. S. Savel’ev, V.A. Yampol’skii, A.L. Rakhmanov, F. Nori, Terahertz Josephson plasma waves in layered superconductors: spectrum, generation, nonlinear and quantum phenomena. Rep. Prog. Phys. 73, 026501 (2010)CrossRefGoogle Scholar
  146. A. Shapira, N. Voloch-Bloch, B.A. Malomed, A. Arie, Spatial quadratic solitons guided by narrow layers of a nonlinear material. J. Opt. Soc. Am. B 28, 1481–1489 (2011)CrossRefGoogle Scholar
  147. X. Shi, B.A. Malomed, F. Ye, X. Chen, Symmetric and asymmetric solitons in a nonlocal nonlinear coupler. Phys. Rev. A 85, 053839 (2012)CrossRefGoogle Scholar
  148. X. Shi, F. Ye, B. Malomed, X. Chen, Nonlinear surface lattice coupler. Opt. Lett. 38, 1064–1066 (2013)CrossRefGoogle Scholar
  149. A. Sigler, B.A. Malomed, Solitary pulses in linearly coupled cubic-quintic Ginzburg-Landau equations. Phys. D 212, 305–316 (2005)CrossRefGoogle Scholar
  150. Y. Silberberg, Collapse of optical pulses. Opt. Lett. 22, 1282–1284 (1990)CrossRefGoogle Scholar
  151. F. Smektala, C. Quemard, V. Couderc, A. Barthélémy, Non-linear optical properties of chalcogenide glasses measured by Z-scan. J. Non-Cryst. Solids 274, 232–237 (2000)CrossRefGoogle Scholar
  152. D.A. Smirnova, A.V. Gorbach, I.V. Iorsh, I.V. Shadrivov, Y.S. Kivshar, Nonlinear switching with a graphene coupler. Phys. Rev. B 88, 045443 (2013)CrossRefGoogle Scholar
  153. N.F. Smyth, A.L. Worthy, Dispersive radiation and nonlinear twin-core fibers. J. Opt. Soc. Am. B 14, 2610–2617 (1997)CrossRefGoogle Scholar
  154. A.W. Snyder, D.J. Mitchell, L. Poladian, D.R. Rowland, Y. Chen, Physics of nonlinear fiber couplers. J. Opt. Soc. Am. B 8, 2102–2112 (1991)CrossRefGoogle Scholar
  155. J.M. Soto-Crespo, N. Akhmediev, Stability of the soliton states in a nonlinear fiber coupler. Phys. Rev. E 48, 4710–4715 (1993)CrossRefGoogle Scholar
  156. J.M. Soto-Crespo, N.N. Akhmediev, V.V. Afanasjev, Stability of the pulselike solutions of the quintic complex Ginzburg–Landau equation. J. Opt. Soc. Am. B 13, 1439–1449 (1996)CrossRefGoogle Scholar
  157. K.E. Strecker, G.B. Partridge, A.G. Truscott, R.G. Hulet, Bright matter wave solitons in Bose-Einstein condensates. New J. Phys. 5, 73.1 (2003)CrossRefGoogle Scholar
  158. S.V. Suchkov, A.A. Sukhorukov, J. Huang, S.V. Dmitriev, C. Lee, Y.S. Kivshar, Nonlinear switching and solitons in \(\mathcal{P}\mathcal{T}\)-symmetric photonic systems. Laser Photon. Rev. 10, 177–213 (2016)Google Scholar
  159. Y. Sun, T.P. White, A.A. Sukhorukov, Coupled-mode theory analysis of optical forces between longitudinally shifted periodic waveguides. J. Opt. Soc. Am. B 30, 736–742 (2013)CrossRefGoogle Scholar
  160. R.S. Tasgal, B.A. Malomed, Modulational instabilities in the dual-core nonlinear optical fiber. Phys. Scr. 60, 418–422 (1999)CrossRefGoogle Scholar
  161. S. Trillo, S. Wabnitz, Coupling instability and power-induced switching with 2-core dual-polarization fiber nonlinear coupler. J. Opt. Soc. Am. B 5, 483–491 (1988)CrossRefGoogle Scholar
  162. S. Trillo, S. Wabnitz, E.M. Wright, G.I. Stegeman, Soliton switching in fiber nonlinear directional couplers. Opt. Lett. 13, 672–674 (1988)CrossRefGoogle Scholar
  163. S. Trillo, G. Stegeman, E. Wright, S. Wabnitz, Parametric amplification and modulational instabilities in dispersive nonlinear directional couplers with relaxing nonlinearity. J. Opt. Soc. Am. B 6, 889–900 (1989)CrossRefGoogle Scholar
  164. S.C. Tsang, K.S. Chiang, K.W. Chow, Soliton interaction in a two-core optical fiber. Opt. Commun. 229, 431–439 (2004)CrossRefGoogle Scholar
  165. Y.J. Tsofe, B.A. Malomed, Quasisymmetric and asymmetric gap solitons in linearly coupled Bragg gratings with a phase shift. Phys. Rev. E 75, 056603 (2007)CrossRefGoogle Scholar
  166. A.V. Ustinov, H. Kohlstedt, M. Cirillo, N.F. Pedersen, G. Hallmanns, G. Heiden, Coupled fluxon modes in stacked Nb/AlOx/Nb long Josephson junctions. Phys. Rev. B 48, 10614–10617 (1993)CrossRefGoogle Scholar
  167. I.M. Uzunov, R. Muschall, M. Gölles, Y.S. Kivshar, B.A. Malomed, F. Lederer, Pulse switching in nonlinear fiber directional couplers. Phys. Rev. E 51, 2527–2537 (1995)CrossRefGoogle Scholar
  168. M. van Hecke, Coherent and incoherent structures in systems described by the 1D CGLE: experiments and identification. Phys. D 174, 134–151 (2003)CrossRefGoogle Scholar
  169. W. van Saarloos, P.C. Hohenberg, Pulses and fronts in the complex Ginzburg-Landau equation near a subcritical bifurcation. Phys. Rev. Lett. 64, 749–752 (1990)CrossRefGoogle Scholar
  170. A. Villeneuve, C.C. Yang, P.C.J. Wigley, G.I. Stegeman, J.S. Aitchison, C.N. Ironside, Ultrafast all-optical switching in semiconductor nonlinear directional couplers at half the band-gap. Appl. Phys. Lett. 61, 147–149 (1992)CrossRefGoogle Scholar
  171. Y.I. Voloshchenko, Y.N. Ryzhov, V.E. Sotin, Stationary waves in nonlinear, periodically modulated media with large group retardation. Zh. Tekh. Fiz. 51, 902–907 (1981) [Sov. Phys. Tech. Phys. 26, 541–544 (1982)]Google Scholar
  172. D.T. Walton, H.G. Winful, Passive mode locking with an active nonlinear directional coupler: positive group-velocity dispersion. Opt. Lett. 18, 720–722 (1993)CrossRefGoogle Scholar
  173. H.G. Winful, D.T. Walton, Passive mode locking through nonlinear coupling in a dual-core fiber laser. Opt. Lett. 17, 1688–1690 (1992)CrossRefGoogle Scholar
  174. E.M. Wright, G.I. Stegeman, S. Wabnitz, Solitary-wave decay and symmetry-breaking instabilities in two-mode fibers. Phys. Rev. A 40, 4455 (1989)CrossRefGoogle Scholar
  175. Y.D. Wu, Coupled-soliton all-optical logic device with two parallel tapered waveguides. Fiber Integr. Opt. 23, 405–414 (2004)CrossRefGoogle Scholar
  176. A. Zafrany, B.A. Malomed, I.M. Merhasin, Solitons in a linearly coupled system with separated dispersion and nonlinearity. Chaos 15, 037108 (2005)CrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Faculty of Engineering, Department of Physical Electronics, School of Electrical EngineeringTel Aviv UniversityTel AvivIsrael
  2. 2.ITMO UniversitySt. PetersburgRussia

Personalised recommendations