Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Frontiers in multidimensional self-trapping of nonlinear fields and matter


2D and 3D solitons and related states, such as quantum droplets, can appear in optical systems, atomic Bose–Einstein condensates (BECs) and liquid crystals, among other physical settings. However, multidimensional solitary states supported by the standard cubic nonlinearity tend to be strongly unstable — a property far less present in 1D systems. Thus, the central challenge is to stabilize multidimensional states, and to that end numerous approaches have been proposed over the years. Most strategies involve non-cubic nonlinearities or using various potentials, including periodic ones. Completely new directions have recently emerged in two-component BECs with spin–orbit coupling, which have been predicted to support stable 2D and metastable 3D solitons. A recent breakthrough is the creation of 3D quantum droplets. These are self-sustained states existing in two-component BECs, stabilized by the quantum fluctuations around the underlying mean-field states. Here, we review recent results in this field and outline outstanding current challenges.

Key points

  • We provide a brief summary of recent theoretical and experimental advances in the study of multidimensional solitons, chiefly in nonlinear optics, ultracold bosonic gases and liquid-crystal and magnetic media.

  • We cover results for fundamental nonlinear modes and for topologically non-trivial states, such as vortex solitons, hopfions and skyrmions.

  • The experimental realization of multidimensional solitons has proved to be more challenging than for 1D solitons owing to the propensity to instabilities of 2D and 3D states, both fundamental and topologically structured. We address different stabilization mechanisms that have been put forward to potentially observe multidimensional solitons, such as competing nonlinearities, linear and nonlinear potentials, spin–orbit coupling, quantum corrections and dissipative effects.

  • Special attention is paid to recent theoretical and experimental results that produced stable 3D solitons in the form of quantum droplets in ultracold bosonic gases, the stability of which is secured by a macroscopic effect of quantum fluctuations around mean-field states.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Observed excitation of short-lived fundamental and vortex light bullets in hexagonal waveguiding arrays.
Fig. 2: Predicted light bullets in radially symmetric and complex potentials.
Fig. 3: Predicted hopfions and hybrid 3D vortex solitons.
Fig. 4: Predicted stable multidimensional solitons in Bose–Einstein condensates with spin–orbit coupling.
Fig. 5: Predicted vortex solitons in the binary Bose–Einstein condensates coupled by a microwave field.
Fig. 6: Observed self-sustained multidimensional quantum droplets.
Fig. 7: Observed topological solitons with differently knotted nematic fields in a liquid crystal.


  1. 1.

    Akhmediev, N. N. & Ankiewicz, A. Solitons: Nonlinear Pulses and Beams (Chapman & Hall, 1997).

  2. 2.

    Kivshar, Y. S. & Agrawal, G. P. Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003).

    Google Scholar 

  3. 3.

    Dauxois, T. & Peyrard, M. Physics of Solitons (Cambridge University Press, Cambridge, 2006).

    MATH  Google Scholar 

  4. 4.

    Malomed, B. A., Mihalache, D., Wise, F. & Torner, L. Spatiotemporal optical solitons. J. Opt. B 7, R53–R72 (2005).

    ADS  Article  Google Scholar 

  5. 5.

    Mihalache, D. Multidimensional localized structures in optical and matter-wave media: atopical review of recent literature. Rom. Rep. Phys. 69, 403 (2017).

    Google Scholar 

  6. 6.

    Radu, E. & Volkov, M. S. Stationary ring solitons in field theory - knots and vortons. Phys. Rep. 468, 101–151 (2008).

    ADS  MathSciNet  Article  Google Scholar 

  7. 7.

    Bender, M., Heenen, P.-H. & Reinhard, P.-G. Self-consistent mean-field models for nuclear structure. Rev. Mod. Phys. 75, 121–180 (2003).

    ADS  Article  Google Scholar 

  8. 8.

    Ruostekoski, J. Stable particlelike solitons with multiply quantized vortex lines in Bose-Einstein condensates. Phys. Rev. A 70, 041601(R) (2004).

    ADS  Article  Google Scholar 

  9. 9.

    Tiurev, K. et al. Three-dimensional skyrmions in spin-2 Bose–Einstein condensates. New J. Phys. 20, 055011 (2018).

    ADS  Article  Google Scholar 

  10. 10.

    Ray, M. W., Ruokokoski, E., Kandel, S., Möttönen, M. & Hall, D. S. Observation of Dirac monopoles in a synthetic magnetic field. Nature 505, 657–660 (2014).

    ADS  Article  Google Scholar 

  11. 11.

    Nguyen, J. H. V., Dyke, P., Luo, D., Malomed, B. A. & Hulet, R. G. Collisions of matter-wave solitons. Nat. Phys. 10, 918–922 (2014).

    Article  Google Scholar 

  12. 12.

    Berge, L. Wave collapse in physics: Principles and applications to light and plasma waves. Phys. Rep. 303, 259–372 (1998).

    ADS  MathSciNet  Article  Google Scholar 

  13. 13.

    Kuznetsov, E. A. & Dias, F. Bifurcations of solitons and their stability. Phys. Rep. 507, 43–105 (2011).

    ADS  MathSciNet  Article  Google Scholar 

  14. 14.

    Fibich, G. The Nonlinear Schrödinger Equation: Singular Solutions and Optical Collapse (Springer, Heidelberg, 2015).

    MATH  Book  Google Scholar 

  15. 15.

    Firth, W. J. & Skryabin, D. V. Optical solitons carrying orbital angular momentum. Phys. Rev. Lett. 79, 2450–2453 (1997).

    ADS  Article  Google Scholar 

  16. 16.

    Torner, L. & Petrov, D. V. Azimuthal instabilities and self-breaking of beams into sets of solitons in bulk second-harmonic generation. Electron. Lett. 33, 608–610 (1997).

    Article  Google Scholar 

  17. 17.

    Silberberg, Y. Collapse of optical pulses. Opt. Lett. 22, 1282–1284 (1990).

    ADS  Article  Google Scholar 

  18. 18.

    Malomed, B. A., Mihalache, D., Wise, F. & Torner, L. Viewpoint: on multidimensional solitons and their legacy in contemporary atomic,molecular and optical physics. J. Phys. B 49, 170502 (2016).

    ADS  Article  Google Scholar 

  19. 19.

    Malomed, B. A. Multidimensional solitons: Well-established results and novel findings. Eur. Phys. J. Spec. Top. 225, 2507–2532 (2016).

    Article  Google Scholar 

  20. 20.

    Lee, T. D., Huang, K. & Yang, C. N. Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties. Phys. Rev. 106, 1135–1145 (1957).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Petrov, D. S. Quantum mechanical stabilization of a collapsing Bose-Bose mixture. Phys. Rev. Lett. 115, 155302 (2015).

    ADS  Article  Google Scholar 

  22. 22.

    Petrov, D. S. & Astrakharchik, G. E. Ultradilute low-dimensional liquids. Phys. Rev. Lett. 117, 100401 (2016).

    ADS  Article  Google Scholar 

  23. 23.

    Baillie, D., Wilson, R. M., Bisset, R. N. & Blakie, P. B. Self-bound dipolar droplet: A localized matter wave in free space. Phys. Rev. A 94, 021602R (2016).

    ADS  Article  Google Scholar 

  24. 24.

    Schmitt, M., Wenzel, M., Böttcher, B., Ferrier-Barbut, I. & Pfau, T. Self-bound droplets of a dilute magnetic quantum liquid. Nature 539, 259–262 (2016).

    ADS  Article  Google Scholar 

  25. 25.

    Ferrier-Barbut, I., Kadau, H., Schmitt, M., Wenzel, M. & Pfau, T. Observation of quantum droplets in a strongly dipolar Bose gas. Phys. Rev. Lett. 116, 215301 (2016).

    ADS  Article  Google Scholar 

  26. 26.

    Chomaz, L. et al. Quantum-fluctuation-driven crossover from a dilute Bose-Einstein condensate to a macrodroplet in a dipolar quantum fluid. Phys. Rev. X 6, 041039 (2016).

    Google Scholar 

  27. 27.

    Cabrera, C. R. et al. Quantum liquid droplets in a mixture of Bose-Einstein condensates. Science 359, 301–304 (2018).

    ADS  MathSciNet  Article  Google Scholar 

  28. 28.

    Cheiney, P. et al. Bright soliton to quantum droplet transition in a mixture of Bose-Einstein condensates. Phys. Rev. Lett. 120, 135301 (2018).

    ADS  Article  Google Scholar 

  29. 29.

    Semeghini, G. et al. Self-bound quantum droplets in atomic mixtures. Phys. Rev. Lett. 120, 235301 (2018).

    ADS  Article  Google Scholar 

  30. 30.

    Li, Y. et al. Two-dimensional solitons and quantum droplets supported by competing self- and cross-interactions in spin-orbit-coupled condensates. New J. Phys. 19, 113043 (2017).

    ADS  Article  Google Scholar 

  31. 31.

    Chiao, R. Y., Garmire, E. & Townes, C. H. Self-trapping of optical beams. Phys. Rev. Lett. 13, 479–482 (1964).

    ADS  Article  Google Scholar 

  32. 32.

    Vakhitov, M. & Kolokolov, A. Stationary solutions of the wave equation in a medium with nonlinearity saturation. Radiophys. Quantum Electron. 16, 783–789 (1973).

    ADS  Article  Google Scholar 

  33. 33.

    Kruglov, V. I. & Vlasov, R. A. Spiral self-trapping propagation of optical beams in media with cubic nonlinearity. Phys. Lett. A 111, 401–404 (1985).

    ADS  Article  Google Scholar 

  34. 34.

    Karamzin, Yu. N. & Sukhorukov, A. P. Mutual focusing of high-power light beams in media with quadratic nonlinearities. Sov. Phys. Jetp. 41, 414–420 (1976).

    ADS  Google Scholar 

  35. 35.

    Stegeman, G. I., Hagan, D. J. & Torner, L. Cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons. Opt. Quantum Electron. 28, 1691–1740 (1996).

    Article  Google Scholar 

  36. 36.

    Etrich, C., Lederer, F., Malomed, B. A., Peschel, T. & Peschel, U. Optical solitons in media with a quadratic nonlinearity. Progress. Opt. 41, 483–568 (2000).

    ADS  Article  Google Scholar 

  37. 37.

    Buryak, A. V., Di Trapani, P., Skryabin, D. V. & Trillo, S. Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications. Phys. Rep. 370, 63–235 (2002).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Mihalache, D. et al. Stable spinning optical solitons in three dimensions. Phys. Rev. Lett. 88, 073902 (2002).

    ADS  Article  Google Scholar 

  39. 39.

    Torruellas, W. E. et al. Observation of two-dimensional spatial solitary waves in a quadratic medium. Phys. Rev. Lett. 74, 5036–5039 (1995).

    ADS  Article  Google Scholar 

  40. 40.

    Petrov, D. V. et al. Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal. Opt. Lett. 23, 1444–1446 (1998).

    ADS  Article  Google Scholar 

  41. 41.

    Kanashov, A. A. & Rubenchik, A. M. On diffraction and dispersion effects on three-wave interaction. Phys. D 4, 122–134 (1981).

    MATH  Article  Google Scholar 

  42. 42.

    Malomed, B. A. et al. Spatio-temporal solitons in optical media with a quadratic nonlinearity. Phys. Rev. E 56, 4725–4735 (1997).

    ADS  Article  Google Scholar 

  43. 43.

    Liu, X., Qian, L. J. & Wise, F. W. Generation of optical spatiotemporal solitons. Phys. Rev. Lett. 82, 4631–4634 (1999).

    ADS  Article  Google Scholar 

  44. 44.

    Liu, X., Beckwitt, K. & Wise, F. W. Two-dimensional optical spatiotemporal solitons in quadratic media. Phys. Rev. E 62, 1328–1340 (2000).

    ADS  Article  Google Scholar 

  45. 45.

    Torner, L., Carrasco, S., Torres, J. P., Crasovan, L.-C. & Mihalache, D. Tandem light bullets. Opt. Commun. 199, 277–281 (2001).

    ADS  Article  Google Scholar 

  46. 46.

    Chen, Z., Segev, M. & Christodoulides, D. N. Optical spatial solitons: historical overview and recent advances. Rep. Prog. Phys. 75, 086401 (2012).

    ADS  Article  Google Scholar 

  47. 47.

    Tikhonenko, V., Christou, J. & Luther-Davies, B. Three-dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium. Phys. Rev. Lett. 76, 2698–2701 (1996).

    ADS  Article  Google Scholar 

  48. 48.

    Bjorkhol, J. & Ashkin, A. CW self-focusing and self-trapping of light in sodium vapour. Phys. Rev. Lett. 32, 129–132 (1974).

    ADS  Article  Google Scholar 

  49. 49.

    Duree, G. C. et al. Observation of self-trapping of an optical beam due to the photorefractive effect. Phys. Rev. Lett. 71, 533–536 (1993).

    ADS  Article  Google Scholar 

  50. 50.

    Iturbe Castillo, M. D., Marquez Aguilar, P. A., Sanchez-Mondragon, J. J., Stepanov, S. & Vysloukh, V. Spatial solitons in photorefractive BTO with drift mechanism of nonlinearity. Appl. Phys. Lett. 64, 408–410 (1994).

    ADS  Article  Google Scholar 

  51. 51.

    Segev, M., Valley, G. C., Crosignani, B., Di Porto, P. & Yariv, A. Steady-state spatial screening solitons in photorefractive materials with external applied-field. Phys. Rev. Lett. 73, 3211–3214 (1994).

    ADS  Article  Google Scholar 

  52. 52.

    Edmundson, D. E. & Enns, R. H. Robust bistable light bullets. Opt. Lett. 17, 586–588 (1992).

    ADS  Article  Google Scholar 

  53. 53.

    Soto-Crespo, J. M., Heatley, D. R., Wright, E. M. & Akhmediev, N. N. Stability of the higher-bound states in a saturable self-focusing medium. Phys. Rev. A 44, 636–644 (1991).

    ADS  Article  Google Scholar 

  54. 54.

    Quiroga-Teixeiro, M. & Michinel, H. Stable azimuthal stationary state in quintic nonlinear media. J. Opt. Soc. Am. B 14, 2004–2009 (1997).

    ADS  Article  Google Scholar 

  55. 55.

    Quiroga-Teixeiro, M. L., Berntson, A. & Michinel, H. Internal dynamics of nonlinear beams in their ground states: short- and long-lived excitation. J. Opt. Soc. Am. B 16, 1697–1704 (1999).

    ADS  Article  Google Scholar 

  56. 56.

    Desyatnikov, A., Maimistov, A. & Malomed, B. Three-dimensional spinning solitons in dispersive media with the cubic-quintic nonlinearity. Phys. Rev. E 61, 3107–3113 (2000).

    ADS  Article  Google Scholar 

  57. 57.

    Paredes, A., Feijoo, D. & Michinel, H. Coherent cavitation in the liquid of light. Phys. Rev. Lett. 112, 173901 (2014).

    ADS  Article  Google Scholar 

  58. 58.

    Falcão-Filho, E. L., de Araújo, C. B., Boudebs, G., Leblond, H. & Skarka, V. Robust two-dimensional spatial solitons in liquid carbon disulfide. Phys. Rev. Lett. 110, 013901 (2013).

    ADS  Article  Google Scholar 

  59. 59.

    Reyna, S., Boudebs, G., Malomed, B. A. & de Araújo, C. B. Robust self-trapping of vortex beams in a saturable optical medium. Phys. Rev. A 93, 013840 (2016).

    ADS  Article  Google Scholar 

  60. 60.

    Reyna, A. S., Jorge, K. C. & de Araújo, C. B. Two-dimensional solitons in a quintic-septimal medium. Phys. Rev. A 90, 063835 (2014).

    ADS  Article  Google Scholar 

  61. 61.

    Akhmediev, N. & Ankiewicz, A. Dissipative Solitons, Lecture Notes in Physics Vol. 661 (Springer, 2005).

  62. 62.

    Grelu, P. & Akhmediev, N. Dissipative solitons for mode-locked lasers. Nat. Photonics 6, 84–92 (2012).

    ADS  Article  Google Scholar 

  63. 63.

    Grelu, P., Soto-Crespo, J. M. & Akhmediev, N. Light bullets and dynamic pattern formation in nonlinear dissipative systems. Opt. Express 13, 9352–9360 (2005).

    ADS  Article  Google Scholar 

  64. 64.

    Skarka, V. & Aleksić, N. B. Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimensional complex cubic-quintic Ginzburg-Landau equations. Phys. Rev. Lett. 96, 013903 (2006).

    ADS  Article  Google Scholar 

  65. 65.

    Mihalache, D. et al. Stable vortex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation. Phys. Rev. Lett. 97, 073904 (2006).

    ADS  Article  Google Scholar 

  66. 66.

    Veretenov, N. A., Rosanov, N. N. & Fedorov, S. V. Rotating and precessing dissipative-optical-topological-3D solitons. Phys. Rev. Lett. 117, 183901 (2016).

    ADS  MathSciNet  Article  Google Scholar 

  67. 67.

    Veretenov, N. A., Fedorov, S. V. & Rosanov, N. N. Topological vortex and knotted dissipative optical 3D solitons generated by 2D vortex solitons. Phys. Rev. Lett. 119, 263901 (2017).

    ADS  Article  Google Scholar 

  68. 68.

    Gustave, F. et al. Observation of mode-locked spatial laser solitons. Phys. Rev. Lett. 118, 044102 (2017).

    ADS  Article  Google Scholar 

  69. 69.

    Bang, O., Krolikowski, W., Wyller, J. & Rasmussen, J. J. Collapse arrest and soliton stabilization in nonlocal nonlinear media. Phys. Rev. E 66, 046619 (2002).

    ADS  MathSciNet  Article  Google Scholar 

  70. 70.

    Conti, C., Peccianti, M. & Assanto, G. Observation of optical spatial solitons in highly nonlocal medium. Phys. Rev. Lett. 92, 113902 (2004).

    ADS  Article  Google Scholar 

  71. 71.

    Rotschild, C., Cohen, O., Manela, O., Segev, M. & Carmon, T. Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons. Phys. Rev. Lett. 95, 213904 (2005).

    ADS  Article  Google Scholar 

  72. 72.

    Burgess, I. B., Peccianti, M., Assanto, G. & Morandotti, R. Accessible light bullets via synergetic nonlinearities. Phys. Rev. Lett. 102, 203903 (2009).

    ADS  Article  Google Scholar 

  73. 73.

    Lahav, O. et al. Three-dimensional spatiotemporal pulse-train solitons. Phys. Rev. X 7, 041051 (2017).

    Google Scholar 

  74. 74.

    Yang, J. & Musslimani, Z. H. Fundamental and vortex solitons in a two-dimensional optical lattice. Opt. Lett. 28, 2094–2096 (2003).

    ADS  Article  Google Scholar 

  75. 75.

    Baizakov, B. B., Malomed, B. A. & Salerno, M. Multidimensional solitons in periodic potentials. Europhys. Lett. 63, 642–648 (2003).

    ADS  MATH  Article  Google Scholar 

  76. 76.

    Efremidis, N. K. et al. Two-dimensional optical lattice solitons. Phys. Rev. Lett. 91, 213906 (2003).

    ADS  Article  Google Scholar 

  77. 77.

    Mihalache, D. et al. Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice. Phys. Rev. E 70, 055603R (2004).

    ADS  MathSciNet  Article  Google Scholar 

  78. 78.

    Baizakov, B. B., Malomed, B. A. & Salerno, M. Multidimensional solitons in a low-dimensional periodic potential. Phys. Rev. A 70, 053613 (2004).

    ADS  Article  Google Scholar 

  79. 79.

    Leblond, H., Malomed, B. A. & Mihalache, D. Three-dimensional vortex solitons in quasi-two-dimensional lattices. Phys. Rev. E 76, 026604 (2007).

    ADS  MathSciNet  Article  Google Scholar 

  80. 80.

    Christodoulides, D. N., Lederer, F. & Silberberg, Y. Discretizing light behavior in linear and nonlinear waveguide lattices. Nature 424, 817–823 (2003).

    ADS  Article  Google Scholar 

  81. 81.

    Fleischer, J. W., Segev, M., Efremidis, N. K. & Christodoulides, D. N. Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices. Nature 422, 147–150 (2003).

    ADS  Article  Google Scholar 

  82. 82.

    Lederer, F. et al. Discrete solitons in optics. Phys. Rep. 463, 1–126 (2008).

    ADS  Article  Google Scholar 

  83. 83.

    Kartashov, Y. V., Vysloukh, V. A. & Torner, L. Soliton shape and mobility control in optical lattices. Progr. Opt. 52, 63–148 (2009).

    Article  Google Scholar 

  84. 84.

    Efremidis, N. K., Sears, S., Christodoulides, D. N., Fleischer, J. W. & Segev, M. Discrete solitons in photorefractive optically induced photonic lattices. Phys. Rev. E 66, 046602 (2002).

    ADS  Article  Google Scholar 

  85. 85.

    Neshev, D. N. et al. Observation of discrete vortex solitons in optically induced photonic lattices. Phys. Rev. Lett. 92, 123903 (2004).

    ADS  Article  Google Scholar 

  86. 86.

    Fleischer, J. W. et al. Observation of vortex-ring discrete solitons in 2D photonic lattices. Phys. Rev. Lett. 92, 123904 (2004).

    ADS  Article  Google Scholar 

  87. 87.

    Terhalle, B. et al. Observation of multivortex solitons in photonic lattices. Phys. Rev. Lett. 101, 013903 (2008).

    ADS  Article  Google Scholar 

  88. 88.

    Chen, Z., Martin, H., Eugenieva, E. D., Xu, J. & Bezryadina, A. Anisotropic enhancement of discrete diffraction and formation of two-dimensional discrete-soliton trains. Phys. Rev. Lett. 92, 143902 (2004).

    ADS  Article  Google Scholar 

  89. 89.

    Martin, H., Eugenieva, E. D., Chen, Z. & Christodoulides, D. N. Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices. Phys. Rev. Lett. 92, 123902 (2004).

    ADS  Article  Google Scholar 

  90. 90.

    Szameit, A. et al. Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica. Opt. Express 14, 6055–6062 (2006).

    ADS  Article  Google Scholar 

  91. 91.

    Szameit, A. & Nolte, S. Discrete optics in femtosecond-laser written photonic structures. J. Phys. B: At. Mol. Opt. Phys. 43, 163001 (2010).

    ADS  Article  Google Scholar 

  92. 92.

    Aceves, A. B., De Angelis, C., Rubenchik, A. M. & Turitsyn, S. K. Multidimensional solitons in fiber arrays. Opt. Lett. 19, 329–331 (1994).

    ADS  Article  Google Scholar 

  93. 93.

    Aceves, A. B., Luther, G. G., De Angelis, C., Rubenchik, A. M. & Turitsyn, S. K. Energy localization in nonlinear fibre arrays: Collapse-effect compressor. Phys. Rev. Lett. 75, 73–76 (1995).

    ADS  Article  Google Scholar 

  94. 94.

    Pelinovsky, D. E. Localization in Periodic Potentials (Cambridge University Press, Cambridge, 2011).

    MATH  Book  Google Scholar 

  95. 95.

    Minardi, S. et al. Three-dimensional light bullets in arrays of waveguides. Phys. Rev. Lett. 105, 263901 (2010).

    ADS  Article  Google Scholar 

  96. 96.

    Eilenberger, F. et al. Evolution dynamics of discrete-continuous light bullets. Phys. Rev. A 84, 013836 (2011).

    ADS  Article  Google Scholar 

  97. 97.

    Eilenberger, F. et al. Observation of discrete, vortex light bullets. Phys. Rev. X 3, 041031 (2013).

    Google Scholar 

  98. 98.

    Mihalache, D. et al. Stable spatiotemporal solitons in Bessel optical lattices. Phys. Rev. Lett. 95, 023902 (2005).

    ADS  Article  Google Scholar 

  99. 99.

    Rüter, C. E. et al. Observation of parity-time symmetry in optics. Nat. Phys. 6, 192–195 (2010).

    Article  Google Scholar 

  100. 100.

    Kartashov, Y. V., Hang, C., Huang, G. X. & Torner, L. Three-dimensional topological solitons in PT-symmetric optical lattices. Optica 3, 1048–1055 (2016).

    Article  Google Scholar 

  101. 101.

    Conti, C. et al. Nonlinear electromagnetic X waves. Phys. Rev. Lett. 90, 170406 (2003).

    ADS  Article  Google Scholar 

  102. 102.

    Di Trapani, P. et al. Spontaneously generated X-shaped light bullets. Phys. Rev. Lett. 91, 093904 (2003).

    ADS  Article  Google Scholar 

  103. 103.

    Lahini, Y. et al. Discrete X-wave formation in nonlinear waveguide arrays. Phys. Rev. Lett. 98, 023901 (2007).

    ADS  Article  Google Scholar 

  104. 104.

    Heinrich, M. et al. Observation of three-dimensional discrete-continuous X waves in photonic lattices. Phys. Rev. Lett. 103, 113903 (2009).

    ADS  Article  Google Scholar 

  105. 105.

    Chong, A., Renninger, W. H., Christodoulides, D. N. & Wise, F. W. Airy-Bessel wave packets as versatile linear light bullets. Nat. Photonics 4, 103–106 (2010).

    ADS  Article  Google Scholar 

  106. 106.

    Abdollahpour, D., Suntsov, S., Papazoglou, D. G. & Tzortzakis, S. Spatiotemporal Airy light bullets in the linear and nonlinear regimes. Phys. Rev. Lett. 105, 253901 (2010).

    ADS  Article  Google Scholar 

  107. 107.

    Agrawal, G. P. Nonlinear Fiber Optics (Academic Press, San Diego, 1995).

    MATH  Google Scholar 

  108. 108.

    Yu, S.-S., Chien, Ch-H., Lai, Y. & Wang, J. Spatio-temporal solitary pulses in graded-index materials with Kerr nonlinearity. Opt. Commun. 119, 167–170 (1995).

    ADS  Article  Google Scholar 

  109. 109.

    Shtyrina, O. V., Fedoruk, M. P., Kivshar, Y. S. & Turitsyn, S. K. Coexistence of collapse and stable spatiotemporal solitons in multimode fibers. Phys. Rev. A 97, 013841 (2018).

    ADS  Article  Google Scholar 

  110. 110.

    Renninger, W. H. & Wise, F. W. Optical solitons in graded-index multimode fibres. Nat. Commun. 4, 1719 (2013).

    ADS  Article  Google Scholar 

  111. 111.

    Wright, L. G., Christodoulides, D. N. & Wise, F. Controllable spatiotemporal nonlinear effects in multimode fibres. Nat. Photonics 9, 306–310 (2015).

    ADS  Article  Google Scholar 

  112. 112.

    Wright, L. G., Wabnitz, S., Christodoulides, D. N. & Wise, F. W. Ultrabroadband dispersive radiation by spatiotemporal oscillation of multimode waves. Phys. Rev. Lett. 115, 223902 (2015).

    ADS  Article  Google Scholar 

  113. 113.

    Wright, L. G. et al. Self-organized instability in graded-index multimode fibres. Nat. Photonics 10, 771–776 (2016).

    ADS  Article  Google Scholar 

  114. 114.

    Krupa, K. et al. Spatial beam self-cleaning in multimode fibres. Nat. Photonics 11, 237–242 (2017).

    ADS  Article  Google Scholar 

  115. 115.

    Kasprzak, J. et al. Bose–Einstein condensation of exciton polaritons. Nature 443, 409–414 (2006).

    ADS  Article  Google Scholar 

  116. 116.

    Amo, A. et al. Polariton superfluids reveal quantum hydrodynamic solitons. Science 332, 1167–1170 (2011).

    ADS  Article  Google Scholar 

  117. 117.

    Nardin, G. et al. Hydrodynamic nucleation of quantized vortex pairs in a polariton quantum fluid. Nat. Phys. 7, 635–641 (2011).

    Article  Google Scholar 

  118. 118.

    Egorov, O. A., Skryabin, D. V., Yulin, A. V. & Lederer, F. Bright cavity polariton solitons. Phys. Rev. Lett. 102, 153904 (2009).

    ADS  Article  Google Scholar 

  119. 119.

    Sich, M. et al. Observation of bright polariton solitons in a semiconductor microcavity. Nat. Photonics 6, 50–55 (2012).

    ADS  Article  Google Scholar 

  120. 120.

    Sala, V. G. et al. Spin-orbit coupling for photons and polaritons in microstructures. Phys. Rev. X 5, 011034 (2015).

    Google Scholar 

  121. 121.

    Schneider, C. et al. Exciton-polariton trapping and potential landscape engineering. Rep. Prog. Phys. 80, 016503 (2017).

    ADS  Article  Google Scholar 

  122. 122.

    Tanese, D. et al. Polariton condensation in solitonic gap states in a one-dimensional periodic potential. Nat. Commun. 4, 1749 (2013).

    Article  Google Scholar 

  123. 123.

    Cerda-Mendez, E. A. et al. Exciton-polariton gap solitons in two-dimensional lattices. Phys. Rev. Lett. 111, 146401 (2013).

    ADS  Article  Google Scholar 

  124. 124.

    Gorbach, A. V., Malomed, B. A. & Skryabin, D. V. Gap polariton solitons. Phys. Lett. A 373, 3024–3027 (2009).

    ADS  MATH  Article  Google Scholar 

  125. 125.

    Ostrovskaya, E. A., Abdullaev, J., Fraser, M. D., Desyatnikov, A. S. & Kivshar, Y. S. Self-localization of polariton condensates in periodic potentials. Phys. Rev. Lett. 110, 170407 (2013).

    ADS  Article  Google Scholar 

  126. 126.

    Nalitov, A. V., Solnyshkov, D. D. & Malpuech, G. Polariton Z topological insulator. Phys. Rev. Lett. 114, 116401 (2015).

    ADS  MathSciNet  Article  Google Scholar 

  127. 127.

    Kartashov, Y. V. & Skryabin, D. V. Modulational instability and solitary waves in polariton topological insulators. Optica 3, 1228–1236 (2016).

    Article  Google Scholar 

  128. 128.

    Tsesses, S. et al. Optical skyrmion lattice in evanescent electromagnetic fields. Science 361, 993–996 (2018).

    ADS  MathSciNet  Article  Google Scholar 

  129. 129.

    Couairon, A. & Mysyrowicz, A. Femtosecond filamentation in transparent media. Phys. Rep. 441, 47–189 (2007).

    ADS  Article  Google Scholar 

  130. 130.

    Durand, M. et al. Self-guided propagation of ultrashort laser pulses in the anomalous dispersion region of transparent solids: a new regime of filamentation. Phys. Rev. Lett. 110, 115003 (2013).

    ADS  Article  Google Scholar 

  131. 131.

    Chekalin, S. V. et al. Light bullets from a femtosecond filament. J. Phys. B 48, 094008 (2015).

    ADS  Article  Google Scholar 

  132. 132.

    Chekalin, S. V., Kompanets, V. O., Dormidonov, A. E. & Kandidov, V. P. Path length and spectrum of single-cycle mid-IR light bullets in transparent dielectrics. Quantum Elec. 48, 372–377 (2018).

    ADS  Article  Google Scholar 

  133. 133.

    Majus, D. et al. Nature of spatiotemporal light bullets in bulk Kerr media. Phys. Rev. Lett. 112, 193901 (2014).

    ADS  Article  Google Scholar 

  134. 134.

    Scheller, M. et al. Externally refueled optical filaments. Nat. Photonics 8, 297 (2014).

    ADS  Article  Google Scholar 

  135. 135.

    Panagiotopoulos, P., Whalen, P., Kolesik, M. & Moloney, J. V. Super high power mid-infrared femtosecond light bullet. Nat. Photonics 9, 543–548 (2015).

    ADS  Article  Google Scholar 

  136. 136.

    Borovkova, O. V., Kartashov, Y. V., Malomed, B. A. & Torner, L. Algebraic bright and vortex solitons in defocusing media. Opt. Lett. 36, 3088–3090 (2011).

    ADS  Article  Google Scholar 

  137. 137.

    Borovkova, O. V., Kartashov, Y. V., Torner, L. & Malomed, B. A. Bright solitons from defocusing nonlinearities. Phys. Rev. E 84, 035602(R) (2011).

    ADS  Article  Google Scholar 

  138. 138.

    Tian, Q., Wu, L., Zhang, Y. & Zhang, J.-F. Vortex solitons in defocusing media with spatially inhomogeneous nonlinearity. Phys. Rev. E 85, 056603 (2012).

    ADS  Article  Google Scholar 

  139. 139.

    Wu, Y., Xie, Q., Zhong, H., Wen, L. & Hai, W. Algebraic bright and vortex solitons in self-defocusing media with spatially inhomogeneous nonlinearity. Phys. Rev. A 87, 055801 (2013).

    ADS  Article  Google Scholar 

  140. 140.

    Driben, R., Kartashov, Y. V., Malomed, B. A., Meier, T. & Torner, L. Soliton gyroscopes in media with spatially growing repulsive nonlinearity. Phys. Rev. Lett. 112, 020404 (2014).

    ADS  Article  Google Scholar 

  141. 141.

    Kartashov, Y. V., Malomed, B. A., Shnir, Y. & Torner, L. Twisted toroidal vortex-solitons in inhomogeneous media with repulsive nonlinearity. Phys. Rev. Lett. 113, 264101 (2014).

    ADS  Article  Google Scholar 

  142. 142.

    Driben, R., Kartashov, Y., Malomed, B. A., Meier, T. & Torner, L. Three-dimensional hybrid vortex solitons. New J. Phys. 16, 063035 (2014).

    ADS  MathSciNet  Article  Google Scholar 

  143. 143.

    Kartashov, Y. V., Malomed, B. A. & Torner, L. Solitons in nonlinear lattices. Rev. Mod. Phys. 83, 247–306 (2011).

    ADS  Article  Google Scholar 

  144. 144.

    Roati, G. et al. 39K Bose-Einstein condensate with tunable interactions. Phys. Rev. Lett. 99, 010403 (2007).

    ADS  Article  Google Scholar 

  145. 145.

    Pollack, S. E. et al. Extreme tunability of interactions in a 7Li Bose-Einstein condensate. Phys. Rev. Lett. 102, 090402 (2009).

    ADS  Article  Google Scholar 

  146. 146.

    Fedichev, P. O., Kagan, Yu, Shlyapnikov, G. V. & Walraven, J. T. M. Influence of nearly resonant light on the scattering length in low-temperature atomic gases. Phys. Rev. Lett. 77, 2913–2916 (1996).

    ADS  Article  Google Scholar 

  147. 147.

    Yan, M., DeSalvo, B. J., Ramachandhran, B., Pu, H. & Killian, T. C. Controlling condensate collapse and expansion with an optical Feshbach resonance. Phys. Rev. Lett. 110, 123201 (2013).

    ADS  Article  Google Scholar 

  148. 148.

    Yamazaki, R., Taie, S., Sugawa, S. & Takahashi, Y. Submicron spatial modulation of an interatomic interaction in a Bose-Einstein condensate. Phys. Rev. Lett. 105, 050405 (2010).

    ADS  Article  Google Scholar 

  149. 149.

    Clark, L. W., Ha, L.-C., Xu, C.-Y. & Chin, C. Quantum dynamics with spatiotemporal control of interactions in a stable Bose-Einstein condensate. Phys. Rev. Lett. 115, 155301 (2015).

    ADS  Article  Google Scholar 

  150. 150.

    Hukriede, J., Runde, D. & Kip, D. Fabrication and application of holographic Bragg gratings in lithium niobate channel waveguides. J. Phys. D 36, R1–R16 (2003).

    ADS  Article  Google Scholar 

  151. 151.

    Morsch, O. & Oberthaler, M. Dynamics of Bose-Einstein condensates in optical lattices. Rev. Mod. Phys. 78, 179–215 (2006).

    ADS  Article  Google Scholar 

  152. 152.

    Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885–964 (2008).

    ADS  Article  Google Scholar 

  153. 153.

    Goldman, N., Budich, J. C. & Zoller, P. Topological quantum matter with ultracold gases in optical lattices. Nat. Phys. 12, 639–645 (2016).

    Article  Google Scholar 

  154. 154.

    Ostrovskaya, E. A. & Kivshar, Y. S. Matter-wave gap solitons in atomic band-gap structures. Phys. Rev. Lett. 90, 160407 (2003).

    ADS  Article  Google Scholar 

  155. 155.

    Louis, P. J. Y., Ostrovskaya, E. A., Savage, C. M. & Kivshar, Y. S. Bose-Einstein condensates in optical lattices: Band-gap structure and solitons. Phys. Rev. A 67, 013602 (2003).

    ADS  Article  Google Scholar 

  156. 156.

    Ostrovskaya, E. A. & Kivshar, Y. S. Matter-wave gap vortices in optical lattices. Phys. Rev. Lett. 93, 160405 (2004).

    ADS  Article  Google Scholar 

  157. 157.

    Sakaguchi, H. & Malomed, B. A. Two-dimensional loosely and tightly bound solitons in optical lattices and inverted traps. J. Phys. B 37, 2225–2239 (2004).

    ADS  Article  Google Scholar 

  158. 158.

    Kartashov, Y. V., Vysloukh, V. A. & Torner, L. Rotary solitons in Bessel optical lattices. Phys. Rev. Lett. 93, 093904 (2004).

    ADS  Article  Google Scholar 

  159. 159.

    Baizakov, B., Malomed, B. A. & Salerno, M. Matter-wave solitons in radially periodic potentials. Phys. Rev. E 74, 066615 (2006).

    ADS  MathSciNet  Article  Google Scholar 

  160. 160.

    Lin, Y.-J., Jiménez-García, K. & Spielman, I. B. Spin-orbit-coupled Bose-Einstein condensates. Nature 471, 83–86 (2011).

    ADS  Article  Google Scholar 

  161. 161.

    Zhai, H. Degenerate quantum gases with spin-orbit coupling: a review. Rep. Prog. Phys. 78, 026001 (2015).

    ADS  MathSciNet  Article  Google Scholar 

  162. 162.

    Sakaguchi, H., Li, B. & Malomed, B. A. Creation of two-dimensional composite solitons in spin-orbit-coupled self-attractive Bose-Einstein condensates in free space. Phys. Rev. E 89, 032920 (2014).

    ADS  Article  Google Scholar 

  163. 163.

    Salasnich, L., Cardoso, W. B. & Malomed, B. A. Localized modes in quasi-two-dimensional Bose-Einstein condensates with spin-orbit and Rabi couplings. Phys. Rev. A 90, 033629 (2014).

    ADS  Article  Google Scholar 

  164. 164.

    Sakaguchi, H., Ya., E., Sherman & Malomed, B. A. Vortex solitons in two-dimensional spin-orbit coupled Bose-Einstein condensates: effects of the Rashba-Dresselhaus coupling and the Zeeman splitting. Phys. Rev. E 94, 032202 (2016).

    ADS  MathSciNet  Article  Google Scholar 

  165. 165.

    Zhang, Y.-C., Zhou, Z.-W., Malomed, B. A. & Pu, H. Stable solitons in three dimensional free space without the ground state: self-trapped Bose-Einstein condensates with spin-orbit coupling. Phys. Rev. Lett. 115, 253902 (2015).

    ADS  Article  Google Scholar 

  166. 166.

    Qin, J., Dong, G. & Malomed, B. A. Stable giant vortex annuli in microwave-coupled atomic condensates. Phys. Rev. A 94, 053611 (2016).

    ADS  Article  Google Scholar 

  167. 167.

    Bulgac, A. Dilute quantum droplets. Phys. Rev. Lett. 89, 050402 (2002).

    ADS  Article  Google Scholar 

  168. 168.

    Volovik, G. E. The Universe in a Helium Droplet (Oxford Univ. Press, 2009).

  169. 169.

    Petrov, D. S. Liquid beyond the van der Waals paradigm. Nat. Phys. 14, 211 (2018).

    Article  Google Scholar 

  170. 170.

    Cappellaro, A., Macrí, T., Bertacco, G. F. & Salasnich, L. Equation of state and self-bound droplet in Rabi-coupled Bose mixtures. Sci. Rep. 7, 13358 (2017).

    ADS  Article  Google Scholar 

  171. 171.

    Cui, X. Spin-orbit coupling induced quantum droplet in ultracold Bose-Fermi mixtures. Phys. Rev. A 98, 023630 (2018).

    ADS  Article  Google Scholar 

  172. 172.

    Saito, H. Path-integral Monte Carlo study on a droplet of a dipolar Bose–Einstein condensate stabilized by quantum fluctuation. J. Phys. Soc. Jpn. 85, 053001 (2016).

    ADS  Article  Google Scholar 

  173. 173.

    Cikojević, V., Dželalija, K., Stipanović, P., Vranješ Markić, L. & Boronat, J. Ul-tradilute quantum liquid drops. Phys. Rev. B 97, 140502R (2018).

    ADS  Article  Google Scholar 

  174. 174.

    Cikojević, V., Vranješ Markić, L., Astrakharchik, G. E. & Boronat, J. Universality in ultradilute liquid Bose-Bose mixtures. Preprint at arXiv (2018).

  175. 175.

    Staudinger, C., Mazzanti, F. & Zillich, R. E. Self-bound Bose mixtures. Phys. Rev. A 98, 023633 (2018).

    ADS  Article  Google Scholar 

  176. 176.

    Li, Y. et al. Two-dimensional vortex quantum droplets. Phys. Rev. A 98, 063602 (2018).

    ADS  Article  Google Scholar 

  177. 177.

    Astrakharchik, G. E. & Malomed, B. A. Dynamics of one-dimensional quantum droplets. Phys. Rev. A 98, 013612 (2018).

    ADS  Article  Google Scholar 

  178. 178.

    Wächtler, F. & Santos, L. Ground-state properties and elementary excitations of quantum droplets in dipolar Bose-Einstein condensates. Phys. Rev. A 94, 043618 (2016).

    ADS  Article  Google Scholar 

  179. 179.

    Xi, K.-T. & Saito, H. Droplet formation in a Bose-Einstein condensate with strong dipole-dipole interaction. Phys. Rev. A 93, 011604R (2016).

    ADS  Article  Google Scholar 

  180. 180.

    Adhikari, S. K. Statics and dynamics of a self-bound dipolar matter-wave droplet. Laser Phys. Lett. 14, 025501 (2017).

    ADS  Article  Google Scholar 

  181. 181.

    Edler, D. et al. Quantum fluctuations in quasi-one-dimensional dipolar Bose-Einstein condensates. Phys. Rev. Lett. 119, 050403 (2017).

    ADS  Article  Google Scholar 

  182. 182.

    Wächtler, F. & Santos, L. Quantum filaments in dipolar Bose-Einstein condensates. Phys. Rev. A 93, 061603R (2016).

    ADS  Article  Google Scholar 

  183. 183.

    Koch, T. et al. Stabilization of a purely dipolar quantum gas against collapse. Nat. Phys. 4, 218–222 (2008).

    Article  Google Scholar 

  184. 184.

    Cidrim, A., dos Santos, F. E. A., Henn, E. A. L. & Macrí, T. Vortices in self-bound dipolar droplets. Phys. Rev. A 98, 023618 (2018).

    ADS  Article  Google Scholar 

  185. 185.

    Kartashov, Y. V., Malomed, B. A., Tarruell, L. & Torner, L. Three-dimensional droplets of swirling superfluids. Phys. Rev. A 98, 013612 (2018).

    ADS  Article  Google Scholar 

  186. 186.

    Ackerman, P. J. & Smalyukh, I. I. Diversity of knot solitons in liquid crystals manifested by linking of preimages in torons and hopfions. Phys. Rev. X 7, 011006 (2017).

    Google Scholar 

  187. 187.

    Ackerman, P. J. & Smalyukh, I. I. Static three-dimensional topological solitons in fluid chiral ferromagnets and colloids. Nat. Mater. 16, 426–432 (2017).

    ADS  Article  Google Scholar 

  188. 188.

    Tai, J.-S., Ackerman, P. I. & Smalyukh, I. I. Topological transformations of Hopf solitons in chiral ferromagnets and liquid crystals. Proc. Nat. Acad. Sci. USA 115, 921–926 (2018).

    ADS  MathSciNet  Article  Google Scholar 

  189. 189.

    Hobart, R. H. On the instability of a class of unitary field models. Proc. Phys. Soc. Lond. 82, 201–203 (1963).

    ADS  MathSciNet  Article  Google Scholar 

  190. 190.

    Derrick, G. H. Comments on nonlinear wave equations as models for elementary particles. J. Math. Phys. 5, 1252–1254 (1964).

    ADS  MathSciNet  Article  Google Scholar 

  191. 191.

    Li, B.-X. et al. Electrically driven three-dimensional solitary waves as director bullets in nematic liquid crystals. Nat. Commun. 9, 2912 (2018).

    ADS  Article  Google Scholar 

  192. 192.

    Lai, P. et al. An improved racetrack structure for transporting a skyrmion. Sci. Rep. 7, 45330 (2017).

    ADS  Article  Google Scholar 

  193. 193.

    Ackerman, P. J., Boyle, T. & Smalyukh, I. I. Squirming motion of baby skyrmions in nematic fluids. Nat. Commun. 8, 673 (2017).

    ADS  Article  Google Scholar 

  194. 194.

    Deng, D.-L., Wang, S.-T., Sun, K. & Duan, L.-M. Probe knots and Hopf insulators with ultracold atoms. Chin. Phys. Lett. 35, 013701 (2018).

    ADS  Article  Google Scholar 

  195. 195.

    Kippenberg, T. J., Gaeta, A. L., Lipson, M. & Gorodetsky, M. L. Dissipative Kerr solitons in optical microcavities. Science 361, 567 (2018).

    Article  Google Scholar 

Download references


The authors greatly appreciate many valuable collaborations and discussions with S. K. Adhikari, G. Dong, A. Gammal, R. G. Hulet, V. V. Konotop, O. D. Lavrentovich, Y. Li, D. Mihalache, D. S. Petrov, H. Sakaguchi, L. Salasnich, E. Y. Sherman, Y. Shnir, D. V. Skryabin and L. Tarruell. L.T. and Y.V.K. acknowledge support from the Severo Ochoa program (SEV-2015-0522) of the Government of Spain, Fundació Cellex, Fundació Mir-Puig, Generalitat de Catalunya and Centres de Recerca de Catalunya (CERCA). The work of B.A.M. is supported, in part, by the joint programme in physics between the US National Science Foundation (NSF) and Binational (US–Israel) Science Foundation through project no. 2015616 and by the Israel Science Foundation through grant no. 1286/17. B.A.M. appreciates the hospitality of the Institute of Photonic Sciences (ICFO) during the preparation of this Review. G.E.A. acknowledges financial support from the Ministry of Science, Innovation and Universities (MICINN, Spain), grant no. FIS2017-84114-C2-1-P.

Author information




All authors contributed to all sections of the paper. 3D solitons in liquid crystals and ferromagnets section was chiefly written by B.A.M.

Corresponding author

Correspondence to Lluis Torner.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.



A class of 3D localized modes, in the form of tori carrying the global vorticity, which are additionally twisted in the torus cross section. This mode carries two independent topological charges (winding numbers), one representing the overall vorticity and the other accounting for the intrinsic twist.


Complex 3D states in various two-component field-theory systems, which carry two independent topological numbers. They were introduced by Skyrme as a classical field model, which effectively describes baryons, and may be derived as a low-energy (semi-classical) limit of quantum chromodynamics.

Spinor Bose–Einstein condensates

(spinor BECs). Condensates composed of two or several components, which may be considered as a set forming a spinor wavefunction, corresponding to a pseudo spin of 1/2 (two components), 1 (three components) or 2 (five components).

Kerr nonlinearities

Universal optical nonlinearities occurring in dielectric media that yield a correction to the local refractive index proportional to the intensity of the electromagnetic wave. They are represented by the cubic self-attractive term in the corresponding nonlinear Schrödinger equation.

Tandem structures

Optical tandem structures represent periodic stacks of materials with different parameters, such as the refractive index, nonlinearity and dispersion, in which widths of individual layers are usually small in comparison with the average diffraction and dispersion lengths.

Vortex solitons

2D or 3D solitons represented by a complex wavefunction whose phase carries an integer winding number (vorticity, also known as the topological charge) and has an amplitude that vanishes at the central pivot.

Parity–time (PT )-symmetry

Special symmetry of the evolution equation or non-Hermitian Hamiltonian governing a dissipative system under the transformation of time reversal and parity inversion (flip of the sign of the spatial coordinate). In the so-called unbroken PT phase, such Hamiltonian shows an entirely real energy spectrum despite being non-Hermitian.


A delocalized linear or nonlinear 2D wave, with the local power featuring an X-shaped profile, which may be supported by defocusing nonlinear optical material when the signs of dispersion and diffraction coefficients are opposite.

Bessel and Airy beams

Bessel beams represent 2D non-diffracting solutions of the Helmholtz equation in circular cylindrical coordinates, in which this equation is separable. Airy beams are non-diffracting 1D or 2D beams that bend along a parabolic trajectory upon propagation while maintaining their functional shapes. Their combinations can be used to construct non-diffracting 3D wavepackets.

Spin–orbit coupling

(SOC). Originally, it referred to the coupling between the spin of electrons in semiconductors and their motion through the crystalline electrostatic field. In the context of the Bose–Einstein condensate, SOC is realized as linear mixing between two components of a binary condensate through first spatial derivatives of the respective wavefunctions.

Topological insulators

Originally, dielectric materials (insulators) possessing a complete gap in the bulk, but admitting conductance through in-gap edge states existing as a result of peculiarities of the intrinsic topological structure of the material. This name is also used for photonic settings that emulate the same phenomenology in terms of light transmission.


Effective potentials that are induced by the nonlinearity whose local strength is subject to spatial modulation.

Feshbach resonances

The effects that make it possible to change the magnitude and sign of the scattering length characterizing collisions between atoms in quantum gases. The Feshbach resonance is a powerful experimental tool enabling control of the strength and sign of the effective nonlinearity in Bose–Einstein condensates (as concerns both self-interactions and cross interactions, in the case of two-component condensates).


A stable two-component 2D or 3D soliton in a two-component spin–orbit-coupled Bose–Einstein condensate, in which, unlike mixed modes, one component has zero vorticity, whereas the other one carries a vorticity of 1.


Stable 2D and 3D solitons in a two-component Bose–Einstein condensate with the spin–orbit coupling between the components. Unlike semi-vortices, each component of such a mode is a mixture of terms with zero vorticity and vorticities of ±1.


Toroidal localized modes that may be created in liquid crystals and ferrofluids. A toron is organized in essentially the same way as a hopfion (a twisted torus, which may carry the overall vorticity).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kartashov, Y.V., Astrakharchik, G.E., Malomed, B.A. et al. Frontiers in multidimensional self-trapping of nonlinear fields and matter. Nat Rev Phys 1, 185–197 (2019).

Download citation

Further reading


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing