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Solitons and collapses: two evolution scenarios of nonlinear wave systems

 a, b, c
a Lebedev Physical Institute, Russian Academy of Sciences, Leninsky prosp. 53, Moscow, 119991, Russian Federation
b Landau Institute for Theoretical Physics, Russian Academy of Sciences, ul. Kosygina 2, Moscow, 119334, Russian Federation
c Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russian Federation

Two alternative scenarios pertaining to the evolution of nonlinear wave systems are considered: solitons and wave collapses. For the former, it suffices that the Hamiltonian be bounded from below (or above), and then the soliton realizing its minimum (or maximum) is Lyapunov stable. The extremum is approached via the radiation of small-amplitude waves, a process absent in systems with finitely many degrees of freedom. The framework of the nonlinear Schrödinger equation and the three-wave system is used to show how the boundedness of the Hamiltonian — and hence the stability of the soliton minimizing it — can be proved rigorously using the integral estimate method based on the Sobolev embedding theorems. Wave systems with the Hamiltonians unbounded from below must evolve to a collapse, which can be considered as the fall of a particle in an unbounded potential. The radiation of small-amplitude waves promotes collapse in this case.

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Fulltext is also available at DOI: 10.3367/UFNe.0182.201206a.0569
PACS: 42.65.Jx, 42.65.Tg, 47.35.Fg, 47.35.Jk, 52.35.Sb (all)
DOI: 10.3367/UFNe.0182.201206a.0569
URL: https://ufn.ru/en/articles/2012/6/a/
Citation: Zakharov V E, Kuznetsov E A "Solitons and collapses: two evolution scenarios of nonlinear wave systems" Phys. Usp. 55 535–556 (2012)
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Received: 14th, July 2011, 2nd, August 2011

Оригинал: Захаров В Е, Кузнецов Е А «Солитоны и коллапсы: два сценария эволюции нелинейных волновых систем» УФН 182 569–592 (2012); DOI: 10.3367/UFNr.0182.201206a.0569

References (103) ↓ Cited by (80) Similar articles (20)

  1. Arnol’d V I Teoriya Katastrof (Novoe v zhizni, nauke, tekhnike. Ser. "Matematika, kibernetika", №9) (M.: Znanie, 1981); Arnold V I Catastrophe Theory (Berlin: Springer, 1986)
  2. Arnol’d V I Matematicheskie Metody Klassicheskoi Mekhaniki 3-e izd. (M.: Nauka, 1989); Arnold V I Mathematical Methods Of Classical Mechanics (New York: Springer, 1997)
  3. Kuznetsov E A, Rubenchik A M, Zakharov V E Phys. Rep. 142 103 (1986)
  4. Kuznetsov E A, Zakharov V E Lecture Notes Phys. 542 3 (2000)
  5. Hasegawa A, Tappet F Appl. Phys. Lett. 23 142 (1973)
  6. Agrawal G P Nonlinear Fiber Optics (New York: Academic Press, 1989); Agraval G Nelineinaya Volokonnaya Optika (M.: Mir, 1996)
  7. Bloembergen N Nonlinear Optics (Reading, Mass.: W.A. Benjamin, 1977)
  8. Zakharov V E, Manakov S V Zh. Eksp. Teor. Fiz. 69 1654 (1975); Zakharov V E, Manakov S V Sov. Phys. JETP 42 842 (1975)
  9. Karamzin Yu N, Sukhorukov A P Zh. Eksp. Teor. Fiz. 68 834 (1975); Karamzin Yu N, Sukhorukov A P Sov. Phys. JETP 41 414 (1975)
  10. Kanashev A A, Rubenchik A M Physica D 4 122 (1981)
  11. Zakharov V E, Kuznetsov E A Usp. Fiz. Nauk 167 1137 (1997); Zakharov V E, Kuznetsov E A Phys. Usp. 40 1087 (1997)
  12. Derrick G J. Math. Phys. 5 1252 (1966)
  13. Benjamin T B Proc. R. Soc. London A 328 153 (1972)
  14. Zakharov V E, Kuznetsov E A Zh. Eksp. Teor. Fiz. 66 594 (1974); Zakharov V E, Kuznetsov E A Sov. Phys. JETP 39 285 (1974)
  15. Krylov S F, Yan’kov V V Zh. Eksp. Teor. Fiz. 79 82 (1980); Krylov S F, Yan’kov V V Sov. Phys. JETP 52 41 (1980)
  16. D’yachenko A I i dr. Zh. Eksp. Teor. Fiz. 96 2026 (1989); D’yachenko A I et al. Sov. Phys. JETP 69 1144 (1989)
  17. Zakharov V E Zh. Eksp. Teor. Fiz. 62 1745 (1972); Zakharov V E Sov. Phys. JETP 35 908 (1972)
  18. Vlasov S N, Petrishchev V A, Talanov V I Izv. Vuzov Radiofizika 14 1353 (1971); Vlasov S N, Petrishchev V A, Talanov V I Radiophys. Quantum Electron. 14 1062 (1971)
  19. Vakhitov N G, Kolokolov A A Izv. Vuzov Radiofizika 16 1020 (1973); Vakhitov N G, Kolokolov A A Radiophys. Quantum Electron. 16 783 (1971)
  20. Askar’yan G A Zh. Eksp. Teor. Fiz. 42 1567 (1962); Askar’yan G A Sov. Phys. JETP 15 1088 (1962)
  21. Talanov V I Pis’ma ZhETF 2 218 (1965); Talanov V I JETP Lett. 2 138 (1965)
  22. Talanov V I Pis’ma ZhETF 11 303 (1970); Talanov V I JETP Lett. 11 199 (1970)
  23. Chiao R Y, Garmire F, Townes C H Phys. Rev. Lett. 13 479 (1964)
  24. Dawes E L, Marburger J H Phys. Rev. 179 862 (1969)
  25. Litvak A G Zh. Eksp. Teor. Fiz. 57 629 (1969); Litvak A G Sov. Phys. JETP 30 344 (1970)
  26. Vlasov S N, Talanov V I Samofokusirovka Voln (N. Novgorod: IPF RAN, 1997)
  27. Zakharov V E Basic Plasma Physics (Handbook of Plasma Physics) Vol. 2 (Eds A A Galeev, R N Sudan) (Amsterdam: North-Holland, 1984) p. 81
  28. Rasmussen J J, Rypdal K Phys. Scripta 33 481 (1986)
  29. Kuznetsov E A Chaos 6 381 (1996)
  30. Sulem C, Sulem P-L The Nonlinear Schrödinger Equation (New York: Springer, 1999)
  31. Gross E P Nuovo Cimento 20 454 (1961)
  32. Pitaevskii L P Zh. Eksp. Teor. Fiz. 40 646 (1961); Pitaevskii L P Sov. Phys. JETP 13 451 (1961)
  33. Kagan Yu M Usp. Fiz. Nauk 178 633 (2008); Kagan Yu M Phys. Usp. 51 603 (2008)
  34. Kagan Yu, Svistunov B V Phys. Rev. Lett. 79 3331 (1997)
  35. Kagan Yu M, Svistunov B V Pis’ma ZhETF 67 495 (1998); Kagan Yu M, Svistunov B V JETP Lett. 67 521 (1998)
  36. Kagan Yu M, Svistunov B V Zh. Eksp. Teor. Fiz. 105 353 (1994); Kagan Yu M, Svistunov B V JETP 78 187 (1994)
  37. Flambaum V, Kuznetsov E Singularities In Fluids, Optics, And Plasmas (Eds R E Caflisch, G C Papanicolaou) (Dordrecht: Kluwer Acad., 1993) p. 197
  38. Berge L, Rasmussen J J Phys. Lett. A 304 136 (2002)
  39. Zakharov V E, Kuznetsov E A Zh. Eksp. Teor. Fiz. 91 1310 (1986); Zakharov V E, Kuznetsov E A Sov. Phys. JETP 64 773 (1986)
  40. Zakharov V E, Shabat A B Zh. Eksp. Teor. Fiz. 61 118 (1971); Zakharov V E, Shabat A B Sov. Phys. JETP 34 62 (1972)
  41. Kivshar Yu S, Agrawal G P Optical Solitons: From Fibers To Photonic Crystals (Amsterdam: Academic Press, 2003); Kivshar’ Yu S, Agraval G P Opticheskie Solitony. Ot Svetovodov k Fotonnym Kristallam (M.: Fizmatlit, 2005)
  42. Segev M, Stegeman G Phys. Today 51 (8) 42 (1998)
  43. Stegeman G, Segev M Science 286 1518 (1998)
  44. Kivshar Yu S Opt. Quantum Electron. 30 571 (1998)
  45. Ladyzhenskaya O A Matematicheskie Voprosy Dinamiki Neszhimaemoi Vyazkoi Zhidkosti (M.: Nauka, 1970); Ladyzhenskaya O A The Mathematical Theory Of Viscous Incompressible Flow (New York: Gordon and Breach, 1969)
  46. Nirenberg L Ann. Scuola Norm. Sup. Pisa 20 733 (1966)
  47. Weinstein M I Commun. Math. Phys. 87 567 (1983)
  48. Gardner C S et al. Phys. Rev. Lett. 19 1095 (1967)
  49. Berge L, Kuznetsov E A, Rasmussen J J Phys. Rev. E 1340 (1996)
  50. Zakharov V E, Kuznetsov E A Zh. Eksp. Teor. Fiz. 113 1892 (1998); Zakharov V E, Kuznetsov E A JETP 86 1035 (1998)
  51. Kuznetsov E A, Dias F Phys. Rep. 507 43 (2011)
  52. Turitsyn S K Pis’ma ZhETF 61 458 (1995); Turitsyn S K JETP Lett. 61 469 (1995)
  53. Berge L et al. Phys. Rev. E 55 3555 (1997)
  54. Kuznetsov E A Optical Solitons: Theoretical Challenges And Industrial Perspectives (Eds V E Zakharov, S Wabnitz) (Berlin: Springer-Verlag, 1999) p. 31
  55. Kuznetsov E A Trudy Nauchnoi Shkoly "Nelineinye Volny — 2010" (Pod red. A V Gaponova-Grekhova, V I Nekorkina) (N. Novgorod: IPF RAN, 2011) p. 205
  56. Grimshaw R H J, Kuznetsov E A, Shapiro E G Physica D 152-153 325 (2001)
  57. Kuznetsov E A, Musher S L Zh. Eksp. Teor. Fiz. 91 1605 (1986); Kuznetsov E A, Musher S L Sov. Phys. JETP 64 947 (1986)
  58. Kuznetsov E A, Mikhailov A V, Shimokhin I A Physica D 87 201 (1995)
  59. Anderson D, Bonnedal M, Lisak M Phys. Fluids 22 1838 (1979)
  60. Anderson D, Lisak M, Berntson A PRAMANA 57 917 (2001)
  61. Vlasov S N, Talanov V I (1984), chastnoe soobshchenie
  62. Kuznetsov E A, Turitsyn S K Phys. Lett. A 112 273 (1985)
  63. Fraiman G M Zh. Eksp. Teor. Fiz. 88 390 (1985); Fraiman G M Sov. Phys. JETP 61 228 (1985)
  64. Zakharov V E, Shvets V F Pis’ma ZhETF 47 227 (1988); Zakharov V E, Shvets V F JETP Lett. 47 275 (1988)
  65. Landman M J et al. Phys. Rev. A 38 3837 (1988)
  66. Smirnov A M, Fraiman G M Physica D 52 2 (1992)
  67. Berge L, Pesme D Phys. Lett. A 166 116 (1992)
  68. Berge L Phys. Rep. 303 259 (1998)
  69. Shrira V I Dokl. Akad. Nauk SSSR 308 732 (1989)
  70. Zhuk V I, Ryzhov O S Dokl. Akad. Nauk SSSR 263 56 (1982); Zhuk V I, Ryzhov O S Sov. Phys. Dokl. 27 177 (1982)
  71. Ryzhov O S Zhurn. Vychisl. Mat. Mat. Fiz. 30 1804 (1990); Ryzhov O S USSR Comput. Math. Math. Phys. 30 (6) 146 (1990)
  72. D’yachenko A I, Kuznetsov E A Pis’ma ZhETF 59 103 (1994); D’yachenko A I, Kuznetsov E A JETP Lett. 39 108 (1994)
  73. Dyachenko A I, Kuznetsov E A Physica D 87 301 (1995)
  74. Kadomtsev B B, Petviashvili V I Dokl. Akad. Nauk SSSR 192 753 (1970); Kadomtsev B B, Petviashvili V I Sov. Phys. Dokl. 15 539 (1970)
  75. Kadomtsev B B Kollektivnye Yavleniya v Plazme (M.: Nauka, 1976); Kadomtsev B B Reviews Of Plasma Physics Vol. 22 (Ed. V D Shafranov) (New York: Kluwer Acad./Plenum Publ., 2001) p. 1
  76. Zakharov V E Pis’ma ZhETF 22 364 (1975); Zakharov V E JETP Lett. 22 172 (1975)
  77. Kuznetsov E A, Turitsyn S K Zh. Eksp. Teor. Fiz. 82 1457 (1982); Kuznetsov E A, Turitsyn S K Sov. Phys. JETP 55 844 (1982)
  78. Abramyan A A, Stepanyants Yu A, Shrira V I Dokl. Akad. Nauk SSSR 327 460 (1992); Abramyan L A, Stepanyants Yu A, Shrira V I Sov. Phys. Dokl. 37 575 (1992)
  79. Kachanov Yu S, Ryzhov O S Sib. Fiz.-tekh. Zhurn. (1) 34 (1992)
  80. Kachanov Y S, Ryzhov O S, Smith F T J. Fluid Mech. 251 273 (1993)
  81. Turitsyn S K Phys. Rev. E 47 R13 (1993)
  82. Kuznetsov E A et al. Physica D 87 273 (1995)
  83. Budneva O B, Zakharov V E, Synakh V S Fizika Plazmy 1 606 (1975); Budneva O B, Zakharov V E, Synakh V S Sov. J. Plasma Phys. 1 335 (1975)
  84. Zakharov V E, Kuznetsov E A, Musher S L Pis’ma ZhETF 41 125 (1985); Zakharov V E, Kuznetsov E A, Musher S L JETP Lett. 41 154 (1985)
  85. Landau L D, Lifshits E M Kvantovaya Mekhanika. Nerelyativistskaya Teoriya (M.: Nauka, 1989); Landau L D, Lifshitz E M Quantum Mechanics. Non-Relativistic Theory (Oxford: Pergamon Press, 1977)
  86. Vlasov S N, Piskunova L V, Talanov V I IV Intern. Workshop on Nonlinear and Turbulent Processes in Physics, Kiev, October 9 - 22, 1989
  87. Zakharov V E, Kosmatov N E, Shvets V F Pis’ma ZhETF 49 431 (1989); Zakharov V E, Kosmatov N E, Shvets V F JETP Lett. 49 492 (1989)
  88. Malkin V M, Shapiro E G Zh. Eksp. Teor. Fiz. 97 183 (1990); Malkin V M, Shapiro E G Sov. Phys. JETP 70 102 (1990)
  89. Kosmatov N E, Shvets V F, Zakharov V E Physica D 52 16 (1991)
  90. Fibich G, Ren W, Wang X-P Phys. Rev. E 67 056603 (2003)
  91. Berge L, Kuznetsov E A, Rasmussen J J Phys. Rev. E 53 R1340 (1996)
  92. Kuznetsov E A Izv. Vuzov Radiofizika 46 342 (2003); Kuznetsov E A Radiophys. Quantum Electron. 46 307 (2003)
  93. Berge L et al. Phys. Rev. Lett. 89 153902 (2002)
  94. Zharova N A, Litvak A G, Mironov V A Pis’ma ZhETF 75 655 (2002); Zharova N A, Litvak A G, Mironov V A JETP Lett. 75 539 (2002)
  95. Zharova N A, Litvak A G, Mironov V A Zh. Eksp. Teor. Fiz. 123 726 (2003); Zharova N A, Litvak A G, Mironov V A JETP 96 643 (2003)
  96. Gurevich V L Kinetika Fononnykh Sistem (M.: Nauka, 1980)
  97. Dias F, Kuznetsov E A Phys. Lett. A 263 98 (1999)
  98. Petviashvili V I Fizika Plazmy 2 450 (1976); Petviashvili V I Sov. J. Plasma Phys. 2 247 (1976)
  99. Nycander J Chaos 4 253 (1994)
  100. Bordag L A et al. Phys. Lett. A 63 205 (1979)
  101. Kuznetsov E A, Musher S L, Shafarenko A V Pis’ma ZhETF 37 204 (1983); Kuznetsov E A, Musher S L, Shafarenko A V JETP Lett. 37 241 (1983)
  102. Turitsyn S K, Fal’kovich G E Zh. Eksp. Teor. Fiz. 89 258 (1985); Turitsyn S K, Fal’kovich G E Sov. Phys. JETP 62 146 (1985)
  103. Balakin A A i dr. Zh. Eksp. Teor. Fiz. 131 408 (2007); Balakin A A et al. JETP 104 363 (2007)

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