Mathematics of Complexity and Dynamical Systems

2011 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Solitons Interactions

  • Tarmo Soomere
Reference work entry
DOI: https://doi.org/10.1007/978-1-4614-1806-1_101

Article Outline

Glossary

Definition of the Subject

Introduction: Key Equations, Milestones, and Methods

Extended Definitions

Elastic Interactions of One‐Dimensional and Line Solitons

Geometry of Oblique Interactions of KP Line Solitons

Soliton Interactions in Laboratory and Nature

Effects in Higher Dimensions

Applications of Line Soliton Interactions

Future Directions

Bibliography

Keywords

Vortex Depression Radar Soliton Coherence 
This is a preview of subscription content, log in to check access

Bibliography

  1. 1.
    Ablowitz MJ, Segur H (1981) Solitons and the inverse scattering transform. SIAM, PhiladelphiaMATHGoogle Scholar
  2. 2.
    Ablowitz MJ, Kaup DJ, Newell AC, Segur H (1974) The inverse scattering transform – Fourier analysis for nonlinear problems. Stud Appl Math 53:249–315MathSciNetGoogle Scholar
  3. 3.
    Aitchison JS, Weiner AM, Silberberg Y, Oliver MK, Jackel JL, Leaird DE, Vogel EM, Smith PWE (1990) Observation of spatial optical solitons in a nonlinear glass wave‐guide. Opt Lett 15(9):471–473Google Scholar
  4. 4.
    Aitchison JS, Silberberg Y, Weiner AM, Leaird DE, Oliver MK, Jackel JL, Vogel EM, Smith PWE (1991) Spatial optical solitons in planar glass wave‐guides. J Opt Soc Am B-Optical Phys 8(6):1290–1297Google Scholar
  5. 5.
    Aitchison JS, Weiner AM, Silberberg Y, Leaird DE, Oliver MK, Jackel JL, Smith PWE (1991) Experimental‐observation of spatial soliton‐interactions. Opt Lett 16(1):15–17Google Scholar
  6. 6.
    Akhmediev N, Krolikowski W, Snyder AW (1998) Partially coherent solitons of variable shape. Phys Rev Lett 81(21):4632–4635Google Scholar
  7. 7.
    Anastassiou C, Segev M, Steiglitz K, Giordmaine JA, Mitchell M, Shih MF, Lan S, Martin J (1999) Energy‐exchange interactions between colliding vector solitons. Phys Rev Lett 83(12):2332–2335Google Scholar
  8. 8.
    Anderson D, Lisak M (1985) Bandwidth limits due to incoherent soliton interaction in optical-fiber communication-systems. Phys Rev A 32(4):2270–2274Google Scholar
  9. 9.
    Apel JR, Ostrovsky LA, Stepanyants YA, Lynch JF (2007) Internal solitons in the ocean. J Acoust Soc Am 121(2):695–722Google Scholar
  10. 10.
    Arnold JM (1998) Varieties of solitons and solitary waves. Opt Quantum Electron 30:631–647Google Scholar
  11. 11.
    Askar'yan GA (1962) Effects of the gradient of strong electromagnetic beam on electrons and atoms. Sov Phys JETP 15(6):1088–1090Google Scholar
  12. 12.
    Baek Y, Schiek R, Stegeman GI, Baumann I, Sohler W (1997) Interactions between one‐dimensional quadratic solitons. Opt Lett 22(20):1550–1552Google Scholar
  13. 13.
    Barthelemy A, Maneuf S, Froehly C (1985) Soliton propagation and self‐confinement of laser‐beams by Kerr optical non‐linearity. Opt Commun 55(3):201–206Google Scholar
  14. 14.
    Belic MR, Stepken A, Kaiser F (1999) Spiraling behavior of photorefractive screening solitons. Phys Rev Lett 82(3):544–547Google Scholar
  15. 15.
    Berger V, Kohlhase S (1976) Mach‐reflection as a diffraction problem. In: Proc 25th Int Conf on Coastal Eng. ASCE, New York, pp 796–814Google Scholar
  16. 16.
    Biondini G, Chakravarty S (2006) Soliton solutions of the Kadomtsev–Petviashvili II equation. J Math Phys 47(3):033514MathSciNetGoogle Scholar
  17. 17.
    Biondini G, Kodama Y (2003) On a family of solutions of the Kadomtsev–Petviashvili equation which also satisfy the Toda lattice hierarchy. J Phys A-Math General 36(42):10519–10536MathSciNetMATHGoogle Scholar
  18. 18.
    Bjorkholm JE, Ashkin A (1974) cw self‐focusing and self‐trapping of light in sodium vapor. Phys Rev Lett 32(4):129–132Google Scholar
  19. 19.
    Busse FH (1994) Convection driven zonal flows and vortices in the major planets. Chaos 4:123–134Google Scholar
  20. 20.
    Buryak AV, Kivshar YS, Shih MF, Segev M (1999) Induced coherence and stable soliton spiraling. Phys Rev Lett 82(1):81–84Google Scholar
  21. 21.
    Carmon T, Anastassiou C, Lan S, Kip D, Musslimani ZH, Segev M, Christodoulides D (2000) Observation of two‐dimensional multimode solitons. Opt Lett 25(15):1113–1115Google Scholar
  22. 22.
    Chen ZG, Segev M, Coskun TH, Christodoulides DN (1996) Observation of incoherently coupled photorefractive spatial soliton pairs. Opt Lett 21(18):1436–1438Google Scholar
  23. 23.
    Chen X-N, Sharma SD (1994) Nonlinear theory of asymmetric motion of a slender ship in a shallow channel. In: Rood EP (ed) 20th Symposium on Naval Hydrodynamics. US Office on Naval Research, Santa Barbara, pp 386–407Google Scholar
  24. 24.
    Chen X-N, Sharma SD (1997) Zero wave resistance for ships moving in shallow channels at supercritical speeds. J Fluid Mech 335:305–321MATHGoogle Scholar
  25. 25.
    Chen X-N, Sharma SD, Stuntz N (2003) Zero wave resistance for ships moving in shallow channels at supercritical speeds. Part 2. Improved theory and model experiment. J Fluid Mech 478:111–124MATHGoogle Scholar
  26. 26.
    Chen X-N, Sharma SD, Stuntz N (2003) Wave reduction by S‑catamaran at supercritical speeds. J Ship Res 47:145–154Google Scholar
  27. 27.
    Chiao RY, Garmire E, Townes CH (1964) Self‐trapping of optical beams. Phys Rev Lett 13(15):479–482Google Scholar
  28. 28.
    Christodoulides DN (1988) Black and white vector solitons in weakly birefringent optical fibers. Phys Lett A 132(8–9):451–452Google Scholar
  29. 29.
    Christodoulides DN, Joseph RI (1988) Vector solitons in birefringent nonlinear dispersive media. Opt Lett 13(1):53–55Google Scholar
  30. 30.
    Christodoulides DN, Singh SR, Carvalho MI, Segev M (1996) Incoherently coupled soliton pairs in biased photorefractive crystals. Appl Phys Lett 68(13):1763–1765Google Scholar
  31. 31.
    Clamond D, Grue J (2002) Interaction between envelope solitons as a model for freak wave formations. Part I: Long time interaction. Comptes Rendus Mecanique 330(8):575–580MATHGoogle Scholar
  32. 32.
    Clamond D, Francius M, Grue J, Kharif C (2006) Long time interaction of envelope solitons and freak wave formations. Eur J Mech B-Fluids 25(5):536–553MathSciNetMATHGoogle Scholar
  33. 33.
    Davis RE, Acrivos A (1967) Solitary internal waves in deep water. J Fluid Mech 29(3):593–608MATHGoogle Scholar
  34. 34.
    Dauxois T, Peyrard M (2006) Physics of solitons. Cambridge University Press, CambridgeMATHGoogle Scholar
  35. 35.
    Desyatnikov AS, Kivshar YS, Torner L (2005) Optical vortices and vortex solitons. Prog Opt 47:291–391Google Scholar
  36. 36.
    Didenkulova II, Zahibo N, Kurkin AA, Levin BV, Pelinovsky EN, Soomere T (2006) Runup of nonlinearly deformed waves on a coast. Doklady Earth Sci 411(8):1241–1243Google Scholar
  37. 37.
    Drazin PG, Johnson RS (1989) Solitons: An introduction. In: Cambridge Texts in Applied Mathematics. Cambridge University Press, CambridgeGoogle Scholar
  38. 38.
    Duree GC, Shultz JL, Salamo GJ, Segev M, Yariv A, Crosignani B, Diporto P, Sharp EJ, Neurgaonkar RR (1993) Observation of self‐trapping of an optical beam due to the photorefractive effect. Phys Rev Lett 71(4):533–536Google Scholar
  39. 39.
    Engelbrecht J, Salupere A (2005) On the problem of periodicity and hidden solitons for the KdV model. Chaos 15:015114Google Scholar
  40. 40.
    Fermi E, Pasta J, Ulam S (1955) Studies of nonlinear problems. I Los Alamos report LA-1940; (1965) In: Segré E (ed) Collected papers of Enrico Fermi. University of Chicago Press, ChicagoGoogle Scholar
  41. 41.
    Firing E, Beardsley RC (1976) The behaviour of a barotropic eddy on a beta‐plane. J Phys Oceanogr 6:57–65Google Scholar
  42. 42.
    Flierl GR, Larichev VD, Mcwilliams JC, Reznik GM (1980) The dynamics of baroclinic and barotropic solitary eddies. Dyn Atmos Oceans 5(1):1–41Google Scholar
  43. 43.
    Folkes PA, Ikezi H, Davis R (1980) Two‐dimensional interaction of ion‐acoustic solitons. Phys Rev Lett 45(11):902–904Google Scholar
  44. 44.
    Fuerst RA, Canva MTG, Baboiu D, Stegeman GI (1997) Properties of type II quadratic solitons excited by imbalanced fundamental waves. Opt Lett 22(23):1748–1750Google Scholar
  45. 45.
    Funakoshi M (1980) Reflection of obliquely incident large‐amplitude solitary wave. J Phys Soc Japan 49:2371–2379Google Scholar
  46. 46.
    Gabl EF, Lonngren KE (1984) On the oblique collision of unequal amplitude ion‐acoustic solitons in a field‐free plasma. Phys Lett A 100:153–155Google Scholar
  47. 47.
    Gardner CS, Greene JM, Kruskal MD, Miura RM (1974) Korteweg–de Vries equations and generalizations: methods for exact solutions. Commun Pure Appl Math 27:97–133MathSciNetMATHGoogle Scholar
  48. 48.
    Gatz S, Herrmann J (1992) Soliton collision and soliton fusion in dispersive materials with a linear and quadratic intensity depending refraction index change. IEEE J Quantum Electron 28(7):1732–1738Google Scholar
  49. 49.
    Galkin VM, Stepanyants YA (1991) On the existence of stationary solitary waves in a rotating fluid. PMM J Appl Math Mech 55(6):939–943MathSciNetGoogle Scholar
  50. 50.
    Gardner CS, Green JM, Kruskal MD, Miura RM (1967) Method for solving the Kortweg–de Vries equation. Phys Rev Lett 19:1095–1097MATHGoogle Scholar
  51. 51.
    Gasparovic RF, Apel JR, Kasischke ES (1988) An overview of the SAR internal wave signature experiment. J Geophys Res‐Oceans 93(C10):12304–12316Google Scholar
  52. 52.
    Gilman OA, Grimshaw R, Stepanyants YA (1996) Dynamics of internal solitary waves in a rotating fluid. Dyn Atmos Oceans 23(1–4):403–411Google Scholar
  53. 53.
    Gordon JP (1983) Interaction forces among solitons in optical fibers. Opt Lett 8(11):596–598Google Scholar
  54. 54.
    Griffits RW, Hopfinger EJ (1986) Experiments with baroclinic vortex pairs in a rotating fluid. J Fluid Mech 173:501–518Google Scholar
  55. 55.
    Griffiths RW, Hopfinger EJ (1987) Coalescing of geostrophic vortices. J Fluid Mech 178:73–97Google Scholar
  56. 56.
    Grimshaw R (2001) Internal solitary waves. In: Grimshaw R (ed) Environmental Stratified Flows. Kluwer, Dordrecht, pp 1–30Google Scholar
  57. 57.
    Gryanik VM, Doronina TN, Olbers DJ, Warncke TH (2000) The theory of three‐dimensional hetons and vortex‐dominated spreading in localized turbulent convection in a fast rotating stratified fluid. J Fluid Mech 423:71–125MathSciNetMATHGoogle Scholar
  58. 58.
    Haragus-Courcelle M, Pego RL (2000) Spatial wave dynamics of steady oblique wave interactions. Physica D 145:207–232MathSciNetMATHGoogle Scholar
  59. 59.
    Hasegawa A (1989) Optical Solitons in Fibers. Springer, BerlinGoogle Scholar
  60. 60.
    Hasegawa A, Tappert F (1973) Transmission of stationary nonlinear optical pulses in dispersive dielectric fiber: II Normal dispersion. Appl Phys Lett 23:171–172Google Scholar
  61. 61.
    Helfrich KR (2007) Decay and return of internal solitary waves with rotation. Phys Fluid 19(2):026601Google Scholar
  62. 62.
    Hirota R (1971) Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys Rev Lett 27:1192–1194Google Scholar
  63. 63.
    Hopfinger EJ, van Heijst GJF (1993) Vortices in rotating fluids. Annu Rev Fluid Mech 25:241–289Google Scholar
  64. 64.
    Ikezi H, Taylor RJ, Baker DR (1970) Formation and interaction of ion‐acoustic solitons. Phys Rev Lett 25(1):11–14Google Scholar
  65. 65.
    Jiankang W, Lee TS, Chu C (2001) Numerical study of wave interaction generated by two ships moving parallely in shallow water. Comput Meth Appl Mech Engrg 190:2099–2110MATHGoogle Scholar
  66. 66.
    Kadomtsev BB, Petviashvili VI (1970) The stability of solitary waves in weakly dispersive media. Dokl Akad Nauk SSSR 192:532–541Google Scholar
  67. 67.
    Kamenkovich et al. (1987) Synoptic eddies in the ocean. Russian version. Gidrometeoizdat, Leningrad, pp 124Google Scholar
  68. 68.
    Kang JU, Stegeman GI, Aitchison JS, Akhmediev N (1996) Observation of Manako vspatial solitons in AlGaAs planar waveguides. Phys Rev Lett 76(20):3699–3702Google Scholar
  69. 69.
    Kanna T, Lakshmanan M (2003) Exact soliton solutions of coupled nonlinear Schrödinger equations: Shape‐changing collisions, logic gates, and partially coherent solitons. Phys Rev E 67(4):046617 Part 2Google Scholar
  70. 70.
    Karamzin YN, Sukhorukov AP (1976) Mutual focusing of high-power light. Sov Phys JETP 41:414–420Google Scholar
  71. 71.
    Katsis C, Akylas TR (1987) On the excitation of long nonlinear water waves by a moving pressure distribution. Part 2. Three-dimensional effects. J Fluid Mech 177:49–65MATHGoogle Scholar
  72. 72.
    Kaup DJ, Newell AC (1978) Solitons as particles, oscillators, and in slowly changing media: A singular pertubation theory. Proc R Soc London A 361(4):413–446MathSciNetGoogle Scholar
  73. 73.
    Kharif C, Pelinovsky N (2003) Physical mechanisms of the rogue wave phenomenon. Eur J Mech B Fluids 22:603–634MathSciNetMATHGoogle Scholar
  74. 74.
    Khitrova G, Gibbs HM, Kawamura Y, Iwamura H, Ikegami T, Sipe JE, Ming L (1993) Spatial solitons in a self‐focusing semiconductor gain medium. Phys Rev Lett 70(7):920–923Google Scholar
  75. 75.
    Kip D, Wesner M, Shandarov V, Moretti P (1998) Observation of bright spatial photorefractive solitons in a planar strontium barium niobate waveguide. Opt Lett 23(12):921–923Google Scholar
  76. 76.
    Kivshar YS, Luther-Davies B (1998) Dark optical solitons: physics and applications. Phys Rep – Rev Sect Phys Lett 298(2–3):81–197Google Scholar
  77. 77.
    Kivshar YS, Malomed BA (1989) Dynamics of solitons in nearly integrable systems. Rev Mod Phys 61(4):768–915Google Scholar
  78. 78.
    Krolikowski W, Holmstrom SA (1997) Fusion and birth of spatial solitons upon collision. Opt Lett 22(6):369–377Google Scholar
  79. 79.
    Krolikowski W, Luther-Davies B, Denz C, Tschudi T (1998) Annihilation of photorefractive solitons. Opt Lett 23(2):97–99Google Scholar
  80. 80.
    Krolikowski W, Akhmediev N, Luther-Davies B (1999) Collision‐induced shape transformations of partially coherent solitons. Phys Rev E 59(4):4654–4658Google Scholar
  81. 81.
    Kurkin AA, Pelinovsky EN (2004) Freak waves: Facts, theory and modelling. Nizhny Novgorod State Technical University, Nizhny Novgorod (in Russian)Google Scholar
  82. 82.
    Lakhsmanan M, Rajasekhar S (2003) Nonlinear dynamics: integrability, chaos and patterns. Springer, BerlinGoogle Scholar
  83. 83.
    Larichev VD, Reznik GM (1976) Strongly non‐linear, two‐dimensional isolated Rossby waves. Oceanology 16:961–967Google Scholar
  84. 84.
    Larichev VD, Reznik GM (1983) Collision of two‐dimensional solitary Rossby waves. Oceanology 23(5):725–734Google Scholar
  85. 85.
    Lee SJ, Grimshaw RHJ (1990) Upstream‐advancing waves generated by three‐dimensional moving disturbances. Phys Fluids A 2:194–201MathSciNetMATHGoogle Scholar
  86. 86.
    Leo G, Assanto G (1997) Collisional interactions of vectorial spatial solitary waves in type II frequency‐doubling media. J Opt Soc Am B-Opt Phys 14(11):3151–3161Google Scholar
  87. 87.
    Li Y, Sclavounos PD (2002) Three‐dimensional nonlinear solitary waves in shallow water generated by an advancing disturbance. J Fluid Mech 470:383–410MathSciNetMATHGoogle Scholar
  88. 88.
    Manakov SV (1974) On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Sov Phys JETP 38:248–253MathSciNetGoogle Scholar
  89. 89.
    Mase H, Memita T, Yuhi M, Kitano T (2002) Stem waves along vertical wall due to random wave incidence. Coast Eng 44:339–350Google Scholar
  90. 90.
    Masuda A, Marubayashi K, Ishibashi M (1990) A laboratory experiment and numerical simulation of an isolated barotropic eddy in a basin with topographic β. J Fluid Mech 213:641–659Google Scholar
  91. 91.
    Melville WK (1980) On the Mach reflection of solitary waves. J Fluid Mech 98:285–297Google Scholar
  92. 92.
    Miles JW (1977) Obliquely interacting solitary waves. J Fluid Mech 79:157–169MathSciNetMATHGoogle Scholar
  93. 93.
    Miles JW (1977) Resonantly interacting solitary waves. J Fluid Mech 79:171–179MathSciNetMATHGoogle Scholar
  94. 94.
    Mitchell M, Segev M (1997) Self‐trapping of incoherent white light. Nature 387(6636):880–883Google Scholar
  95. 95.
    Mitchell M, Chen ZG, Shih MF, Segev M (1996) Self‐trapping of partially spatially incoherent light. Phys Rev Lett 77(3):490–493Google Scholar
  96. 96.
    Mitchell M, Segev M, Christodoulides DN (1998) Observation of multihump multimode solitons. Phys Rev Lett 80(21):4657–4660Google Scholar
  97. 97.
    Mollenauer LF, Stolen RH, Gordon JP (1980) Experimental‐observation of picosecond pulse narrowing and solitons in optical fibers. Phys Rev Lett 45(13):1095–1098Google Scholar
  98. 98.
    Musslimani ZH, Segev M, Christodoulides DN, Soljacic M (2000) Composite multihump vector solitons carrying topological charge. Phys Rev Lett 84(6):1164–1167Google Scholar
  99. 99.
    Musslimani ZH, Soljacic M, Segev M, Christodoulides DN (2001) Interactions between two‐dimensional composite vector solitons carrying topological charges. Phys Rev E 63(6):066608Google Scholar
  100. 100.
    Musslimani ZH, Soljacic M, Segev M, Christodoulides DN (2001) Delayed‐action interaction and spin‐orbit coupling between solitons. Phys Rev Lett 86(5):799–802Google Scholar
  101. 101.
    Newell AC, Redekopp LG (1977) Breakdown of Zakharov–Shabat theory and soliton creation. Phys Rev Lett 38(8):377–380Google Scholar
  102. 102.
    Nezlin VM, Sneshkin EN (1993) Rossby vortices, spiral structures, solitons: Astrophysics and plasma physics in shallow water experiments. Springer, BerlinGoogle Scholar
  103. 103.
    Nishida Y, Nagasawa T (1980) Oblique collision of plane ion‐acoustic solitons. Phys Rev Lett 45(20):1626–1629Google Scholar
  104. 104.
    Nycander J (1993) The difference between monopole vortices in planetary flows and laboratory experiments. J Fluid Mech 254:561–577MathSciNetMATHGoogle Scholar
  105. 105.
    Oikawa M, Tsuji H (2006) Oblique interactions of weakly nonlinear long waves in dispersive systems. Fluid Dyn Res 38:868–898MathSciNetMATHGoogle Scholar
  106. 106.
    Osborne AR, Bergamasco L (1986) The solitons of Zabusky and Kruskal revisited: perspective in terms of the periodic spectral transform. Physica D 18:26–46MathSciNetMATHGoogle Scholar
  107. 107.
    Osborne AR, Burch TL (1980) Internal solitons in the Andaman Sea. Science 208(4443):451–460Google Scholar
  108. 108.
    Osborne AR, Onorato M, Serio M (2000) The nonlinear dynamics of rogue waves and holes in deep‐water gravity wave trains. Phys Lett A 275(5–6):386–393MathSciNetMATHGoogle Scholar
  109. 109.
    Ostrovskaya EA, Kivshar YS, Chen ZG, Segev M (1999) Interaction between vector solitons and solitonic gluons. Opt Lett 24(5):327–329Google Scholar
  110. 110.
    Ostrovskaya EA, Kivshar YS, Skryabin DV, Firth WJ (1999) Stability of multihump optical solitons. Phys Rev Lett 83(2):296–299Google Scholar
  111. 111.
    Ostrovsky LA, Stepanyants YA (1989) Do internal solitons exist in the ocean? Rev Geophys 27:293–310Google Scholar
  112. 112.
    Parnell KE, Kofoed-Hansen H (2001) Wakes from large high‐speed ferries in confined coastal waters: Management approaches with examples from New Zealand and Denmark. Coastal Manage 29:217–237Google Scholar
  113. 113.
    Pedersen G (1988) Three‐dimensional wave patterns generated by moving disturbances at transcritical speeds. J Fluid Mech 196:39–63MATHGoogle Scholar
  114. 114.
    Pedersen NF, Samuelsen MR, Welner D (1984) Soliton annihilation in the perturbed sine‐Gordon system. Phys Rev B 30(7):4057–4059Google Scholar
  115. 115.
    Peregrine DH (1983) Wave jumps and caustics in the propagation of finite‐amplitude water waves. J Fluid Mech 136:435–452Google Scholar
  116. 116.
    Perring JK, Skyrme THR (1962) A Model Unified Field Equation. Nucl Phys 31:550–555MathSciNetMATHGoogle Scholar
  117. 117.
    Perroud PH (1957) The Solitary Wave Reflection Along a Straight Vertical Wall at Oblique Incidence. Univ. of Calif. Berkeley, IER Rept 99-3, pp 93Google Scholar
  118. 118.
    Peterson P, van Groesen E (2000) A direct and inverse problem for wave crests modelled by interactions of two solitons. Physica D 141:316–332MathSciNetMATHGoogle Scholar
  119. 119.
    Peterson P, van Groesen E (2001) Sensitivity of the inverse wave crest problem. Wave Motion 34:391–399MATHGoogle Scholar
  120. 120.
    Peterson P, Soomere T, Engelbrecht J, van Groesen E (2003) Soliton interaction as a possible model for extreme waves in shallow water. Nonlinear Process Geophys 10:503–510Google Scholar
  121. 121.
    Petviashvili VI (1980) Red Spot of Jupiter and the drift soliton in plasma. JETP Lett 32:619–622Google Scholar
  122. 122.
    Pigier C, Uzdin R, Carmon T, Segev M, Nepomnyaschchy A, Musslimani ZH (2001) Collisions between (\({2+1}\))D rotating propeller solitons. Opt Lett 26(20):1577–1579Google Scholar
  123. 123.
    Poladian L, Snyder AW, Mitchell DJ (1991) Spiraling spatial solitons. Opt Commun 85(1):59–62Google Scholar
  124. 124.
    Porubov AV, Tsuji H, Lavrenov IV, Oikawa M (2005) Formation of the rogue wave due to non‐linear two‐dimensional waves interaction. Wave Motion 42:202–210MathSciNetMATHGoogle Scholar
  125. 125.
    Radhakrishnan R, Lakshmanan M, Hietarinta J (1997) Inelastic collision and switching of coupled bright solitons in optical fibers. Phys Rev E 56(2):2213–2216Google Scholar
  126. 126.
    Read PL, Hide R (1983) Long‐lived eddies in the laboratory and in the atmosphere of Jupiter and Saturn. Nature 302(10):126–129Google Scholar
  127. 127.
    Rebbi C (1979) Solitons. Sci Am 240(2):76–91MathSciNetGoogle Scholar
  128. 128.
    Redekopp LG, Weidman PD (1978) Solitary Rossby waves in zonal shear flows and their interactions. J Atm Sci 35:790–804Google Scholar
  129. 129.
    Rotschild C, Alfassi B, Cohen O, Segev M (2006) Long‐range interactions between optical solitons. Nat Phys 2(11):769–774Google Scholar
  130. 130.
    Rottman JW, Grimshaw R (2001) Atmospheric internal solitary waves. In: Grimshaw R (ed), Environmental Stratified Flows. Kluwer, Dordrecht, pp 91–129Google Scholar
  131. 131.
    Russell JS (1837) Applications and illustrations of the law of wave in the practical navigation of canals. Trans R Soc Edin 14:33–34Google Scholar
  132. 132.
    Russell JS (1844) Report on waves. In: Report of the 14th Meeting of the British Association for the Advancement of Science. Murray, York, pp 311–390Google Scholar
  133. 133.
    Salupere A, Peterson P, Engelbrecht J (2002) Long‐time behaviour of soliton ensembles. Part I – Emergence of ensembles. Chaos Solitons Fractals 14(9):1413–1424MathSciNetMATHGoogle Scholar
  134. 134.
    Salupere A, Peterson P, Engelbrecht J (2003) Long‐time behaviour of soliton ensembles. Part II – Periodical patterns of trajectories. Chaos Solitons Fractals 15:29–40MathSciNetGoogle Scholar
  135. 135.
    Segev M (1998) Optical spatial solitons. Opt Quantum Electron 30(7–10):03–533Google Scholar
  136. 136.
    Segev M, Stegeman G (1998) Self‐trapping of optical beams: Spatial solitons. Phys Today 51(8):42–48Google Scholar
  137. 137.
    Segev M, Crosignani B, Yariv A, Fischer B (1992) Spatial solitons in photorefractive media. Phys Rev Lett 68(7):923–926Google Scholar
  138. 138.
    Shalaby M, Barthelemy AJ (1992) Observation of the self‐guided propagation of a dark and bright spatial soliton pair in a focusing nonlinear medium. IEEE J Quantum Electron 28(12):2736–2741Google Scholar
  139. 139.
    Shalaby M, Reynaud F, Barthelemy A (1992) Experimental‐observation of spatial soliton‐interactions with a pi-2 relative phase difference. Opt Lett 17(11):778–780Google Scholar
  140. 140.
    Shih MF, Segev M (1996) Incoherent collisions between two‐dimensional bright steady‐state photorefractive spatial screening solitons. Opt Lett 21(19):1538–1540Google Scholar
  141. 141.
    Shih MF, Segev M, Salamo G (1997) Three‐dimensional spiraling of interacting spatial solitons. Phys Rev Lett 78(13):2551–2554Google Scholar
  142. 142.
    Snyder AW, Sheppard AP (1993) Collisions, steering, and guidance with spatial solitons. Opt Lett 18(7):482–484Google Scholar
  143. 143.
    Soljacic M, Sears S, Segev M (1998) Self‐trapping of “necklace” beams in self‐focusing Kerr media. Phys Rev Lett 81(22):4851–4854Google Scholar
  144. 144.
    Sommeria J, Nore C, Dumont T, Robert R (1991) Statistical theory of the Great Red Spot of Jupiter. C R Acad Sci II 312:999–1005Google Scholar
  145. 145.
    Soomere T (1992) Geometry of the double resonance of Rossby waves. Ann Geophys 10:741–748Google Scholar
  146. 146.
    Soomere T (2004) Interaction of Kadomtsev–Petviashvili solitons with unequal amplitudes. Phys Lett A 332:74–81MathSciNetMATHGoogle Scholar
  147. 147.
    Soomere T (2005) Fast ferry traffic as a qualitatively new forcing factor of environmental processes in non-tidal sea areas: a case study in Tallinn Bay, Baltic Sea. Environ Fluid Mech 5:4 293–323Google Scholar
  148. 148.
    Soomere T (2006) Nonlinear ship wake waves as a model of rogue waves and a source of danger to the coastal environment: a review. Oceanologia 48(S):185–202Google Scholar
  149. 149.
    Soomere T (2007) Nonlinear components of ship wake waves. Appl Mech Rev 60(3):120–138Google Scholar
  150. 150.
    Soomere T, Engelbrecht J (2005) Extreme elevations and slopes of interacting solitons in shallow water. Wave Motion 41:179–192MathSciNetMATHGoogle Scholar
  151. 151.
    Soomere T, Engelbrecht J (2006) Weakly two-dimensional interaction of solitons in shallow water. Eur J Mech B Fluids 25(5):636–648MathSciNetMATHGoogle Scholar
  152. 152.
    Stegeman GI, Segev M (1999) Optical spatial solitons and their interactions: Universality and diversity. Science 286(5444):1518–1523. doi:10.1126/science.286.5444.1518Google Scholar
  153. 153.
    Stegner A, Zeitlin V (1996) Asymptotic expansions and monopolar solitary Rossby vortices in barotropic and two‐layer model. Geophys Astrophys Fluid Dyn 83:159–195Google Scholar
  154. 154.
    Stegner A, Zeitlin V (1998) From quasi‐geostrophic to strongly non‐linear monopolar vortices in a paraboloidal shallow‐water experiment. J Fluid Mech 356:1–24MathSciNetGoogle Scholar
  155. 155.
    Steiglitz K (2001) Time‐gated Manakov spatial solitons are computationally. Phys Rev E 63(1):016608Google Scholar
  156. 156.
    Stepken A, Kaiser F, Belic MR, Krolikowski W (1998) Interaction of incoherent two‐dimensional photorefractive solitons. Phys Rev E 58(4):R4112–R4115Google Scholar
  157. 157.
    Stern ME (1975) Minimal properties of planetary eddies. J Mar Res 33:1–13Google Scholar
  158. 158.
    Taijiri M, Maesono H (1997) Resonant interactions of drift vortex solitons in a convective motion of a plasma. Phys Rev E 55(3):3351–3357Google Scholar
  159. 159.
    Tanaka M (1993) Mach reflection of a large‐amplitude solitary wave. J Fluid Mech 248:637–661MATHGoogle Scholar
  160. 160.
    Tikhonenko V, Christou J, Luther-Davies B (1995) Spiraling bright spatial solitons formed by the breakup of an optical vortex in a saturable self‐focusing medium. J Opt Soc Am B-Opt Phys 12(11):2046–2052Google Scholar
  161. 161.
    Tikhonenko V, Christou J, Luther-Davies B (1996) Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium. Phys Rev Lett 76(15):2698–2701Google Scholar
  162. 162.
    Toffoli A, Lefevre JM, Bitner-Gregersen E, Monbaliu J (2005) Towards the identification of warning criteria: Analysis of a ship accident database. Appl Ocean Res 27(6):281–291Google Scholar
  163. 163.
    Torner L, Torres JP, Menyuk CR (1996) Fission and self‐deflection of spatial solitons by cascading. Opt Lett 21(7):462–464Google Scholar
  164. 164.
    Torruellas WE, Wang Z, Hagan DJ, Vanstryland EW, Stegeman GI, Torner L, Menyuk CR (1995) Observation of 2‑dimensional spatial solitary waves in a quadratic medium. Phys Rev Lett 74(25):5036–5039Google Scholar
  165. 165.
    Tratnik MV, Sipe JE (1988) Bound solitary waves in a birefringent optical fiber. Phys Rev A 38(4):2011–2017Google Scholar
  166. 166.
    Trillo S, Wabnitz S, Wright EM, Stegeman GI (1988) Optical solitary waves induced by cross‐phase modulation. Opt Lett 13(10):871–873Google Scholar
  167. 167.
    Tsuji H, Oikawa M (2001) Oblique interaction of internal solitary waves in a two‐layer fluid of infinite depth. Fluid Dyn Res 29:251–267MathSciNetMATHGoogle Scholar
  168. 168.
    Tsuji H, Oikawa M (2004) Two‐dimensional interaction of solitary waves in a modified Kadomtsev–Petviashvili equation. J Phys Soc Japan 73:3034–3043MATHGoogle Scholar
  169. 169.
    Washimi H, Taniuti T (1966) Propagation of ion‐acoustic solitary waves of small amplitude. Phys Rev Lett 17:996–998Google Scholar
  170. 170.
    Williams GP, Yamagata T (1984) Geostrophic regimes, intermediate solitary vortices and Jovian eddies. J Atm Sci 41:453–478Google Scholar
  171. 171.
    Yajima N, Oikawa M, Satsuma J (1978) Interaction of ion‐acoustic solitons in three‐dimensional space. J Phys Soc Japan 44:1711–1714Google Scholar
  172. 172.
    Yoon SB, Liu PLF (1989) Stem waves along breakwater. J Waterw Port Coast Ocean Eng – ASCE 115:635–648Google Scholar
  173. 173.
    Yue DK, Mei CC (1980) Forward diffrection of Stokes waves by a thin wedge. J Fluid Mech 99:33–52MathSciNetMATHGoogle Scholar
  174. 174.
    Zabusky NJ, Kruskal MD (1965) Interaction on “solitons” in a collisionless plasma and the recurrence of initial states. Phys Rev Lett 15(6):240–243MATHGoogle Scholar
  175. 175.
    Zakharov VE (1968) Stability of periodic waves of finite amplitude on the surface of a deep fluid. Zh Prikl Mekh Tekh Fiz 9:86–94; J Appl Mech Tech Phys 9:190–194Google Scholar
  176. 176.
    Zakharov VE, Shabat AB (1972) Exact theory of two‐dimensional self‐focusing and one‐dimensional self‐modulation of waves in nonlinear media. Sov Phys JETP 34:62–69MathSciNetGoogle Scholar
  177. 177.
    Zakharov VE, Manakov SV, Novikov SP, Pitaevsky LP (1980) Theory of solitons. Nauka, Moscow (in Russian) (English translation, 1984, Consultants Bureau, New York)MATHGoogle Scholar
  178. 178.
    Ze F, Hershkowitz N, Chan C, Lonngren KE (1979) Inelastic collision of spherical ion‐acoustic solitons. Phys Rev Lett 42(26):1747–1750Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Tarmo Soomere
    • 1
  1. 1.Center for Nonlinear Studies, Institute of CyberneticsTallinn University of TechnologyTallinnEstonia