Issues

 / 

2012

 / 

June

  

Reviews of topical problems


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.

Fulltext pdf (683 KB)
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/
000308868100001
2012PhyU...55..535Z
Citation: Zakharov V E, Kuznetsov E A "Solitons and collapses: two evolution scenarios of nonlinear wave systems" Phys. Usp. 55 535–556 (2012)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

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 (133) ↓ Similar articles (20)

  1. Yao X, Wang L Nonlinear Dyn 113 11907 (2025)
  2. Flamarion M V, Pelinovsky E, Didenkulova E Phys. Wave Phen. 33 9 (2025)
  3. Lashkin V M, Cheremnykh O K 32 (2) (2025)
  4. Chen Sh, Wang L et al Chaos, Solitons & Fractals 190 115777 (2025)
  5. Ruban V P Pisʹma V žurnal êksperimentalʹnoj I Teoretičeskoj Fiziki 120 745 (2024)
  6. Jin X-W, Yang Zh-Y et al Phys. Rev. B 109 (13) (2024)
  7. Dong L, Fan M, Malomed B A Chaos, Solitons & Fractals 188 115499 (2024)
  8. Lashkin V M 31 (4) (2024)
  9. Ruban V P Jetp Lett. 120 713 (2024)
  10. Lashkin V M Phys. Rev. E 109 (6) (2024)
  11. Lashkin V M, Cheremnykh O K 36 (2) (2024)
  12. Zhong M, Chen Y et al Proc. R. Soc. A. 480 (2282) (2024)
  13. Yao X, Ma J, Meng G Nonlinear Dyn 112 18435 (2024)
  14. Ostrovsky L, Pelinovsky E et al 34 (6) (2024)
  15. Malomed B A Advances In Physics: X 9 (1) (2024)
  16. Kochurin E  A, Kuznetsov E  A Phys. Rev. Lett. 133 (20) (2024)
  17. Yu K M, I K K et al Springer Series In Solid-State Sciences Vol. Electronic Phase Separation in Magnetic and Superconducting MaterialsDroplets Formation, BEC and Superconductivity in Quantum Gases, Metallic Hydrogen and Excitonic Systems201 Chapter 14 (2024) p. 289
  18. Liu D, Gao Ya et al Optics & Laser Technology 177 111181 (2024)
  19. Elkamash I S, Reville B et al Chaos, Solitons & Fractals 188 115531 (2024)
  20. Ruban V P Pisʹma V žurnal êksperimentalʹnoj I Teoretičeskoj Fiziki 119 579 (2024)
  21. Ruban V P Jetp Lett. 119 585 (2024)
  22. Gelash A, Dremov S et al Phys. Rev. Lett. 132 (13) (2024)
  23. Lashkin V M, Cheremnykh O K Phys. Rev. E 110 (2) (2024)
  24. Jin X-W, Yang Zh-Y et al Phys. Rev. B 109 (1) (2024)
  25. Yu K M, I K K et al Springer Series In Solid-State Sciences Vol. Electronic Phase Separation in Magnetic and Superconducting MaterialsIntroduction. Spontaneously Formed Nanoscale Inhomogenieties in Different Materials201 Chapter 1 (2024) p. 1
  26. Dong L, Fan M Chaos, Solitons & Fractals 173 113728 (2023)
  27. Li Ch, Konotop V V et al Chaos, Solitons & Fractals 174 113848 (2023)
  28. Kagan M Yu, Aksenov S V et al Pisʹma V žurnal êksperimentalʹnoj I Teoretičeskoj Fiziki 117 754 (2023)
  29. Zemlyanov A A, Minina O V Atmos Ocean Opt 36 314 (2023)
  30. Zemlyanov A A, Minina O V et al XVI International Conference on Pulsed Lasers and Laser Applications, (2023) p. 16
  31. Kukushkin A B, Kulichenko A A Foundations 3 602 (2023)
  32. Chen Zh, Li Y et al Communications In Nonlinear Science And Numerical Simulation 118 107013 (2023)
  33. Kagan M Yu, Aksenov S V et al Jetp Lett. 117 755 (2023)
  34. Kumar Sh, Li P, Malomed B A Phys. Rev. E 108 (2) (2023)
  35. Tribelsky M I Proc. R. Soc. A. 479 (2277) (2023)
  36. Kuznetsov E A Radiophys Quantum El 66 305 (2023)
  37. Lashkin V M, Cheremnykh O K et al Phys. Rev. E 107 (2) (2023)
  38. Levkov D  G, Maslov V  E Phys. Rev. D 108 (6) (2023)
  39. Mezentsev V K, Podivilov E et al Phys. Rev. E 106 (5) (2022)
  40. Malomed B A 48 856 (2022)
  41. Agafontsev D S, Kuznetsov E A et al Phys.-Usp. 65 189 (2022)
  42. Fokas A S, Cao Yu, He J Fractal Fract 6 425 (2022)
  43. Kuznetsov E A J. Exp. Theor. Phys. 135 121 (2022)
  44. Malomed B A Multidimensional Solitons (2022) p. 1-1
  45. Garani R, Levkov D, Tinyakov P Phys. Rev. D 105 (6) (2022)
  46. Vitanov N K Entropy 24 1653 (2022)
  47. Chen Zh, Li Y et al SSRN Journal (2022)
  48. Levkov D G, Maslov V E et al J. High Energ. Phys. 2022 (12) (2022)
  49. Rao J, He J, Malomed B A 63 (1) (2022)
  50. Bica I, Mucalica A 30 45 (2022)
  51. Kontorovich V M, Poslavskyi S A 48 413 (2022)
  52. Malomed B A Multidimensional Solitons (2022) p. 9-1
  53. Deng D, Yuan Zh et al Geophysical Research Letters 49 (4) (2022)
  54. Kuznetsov E A, Kagan M Yu J. Exp. Theor. Phys. 132 704 (2021)
  55. Sinkevich O A High Temp 59 77 (2021)
  56. Kochurin E A, Zubarev N M Fluids 6 125 (2021)
  57. PELAP François Beceau, NDECFO Jean Emac, DEFFO Guy Roger Phys. Scr. 96 075211 (2021)
  58. Khalili S, Hasanbeigi A, Sobhanian S Plasma Phys. Rep. 47 298 (2021)
  59. Belashov V Yu, Kharshiladze O A, Belashova E S Geomagn. Aeron. 61 149 (2021)
  60. Guo L, Chabchoub A, He J Physica D: Nonlinear Phenomena 426 132990 (2021)
  61. Rao J, Chow K W et al Stud Appl Math 147 1007 (2021)
  62. Mullyadzhanov R I, Gelash A A Radiophys Quantum El 63 786 (2021)
  63. Fonkoua S A T, Pelap F B et al Eur. Phys. J. Plus 136 (4) (2021)
  64. Dmitriev A  S, Levkov D  G et al Phys. Rev. D 104 (2) (2021)
  65. Zuev L B Multiscale Biomechanics and Tribology of Inorganic and Organic Systems Springer Tracts In Mechanical Engineering Chapter 12 (2021) p. 245
  66. Alfimov G L, Fedotov A P, Sinelshchikov D I Physica D: Nonlinear Phenomena 402 132245 (2020)
  67. Oloo J O, Shrira V I Teoreticheskaya Matematicheskaya Fizika 203 91 (2020) [Oloo J O, Shrira V I Theor Math Phys 203 512 (2020)]
  68. Chavanis P-H Phys. Rev. D 102 (8) (2020)
  69. Nugaev E Ya, Shkerin A V J. Exp. Theor. Phys. 130 301 (2020)
  70. Kuznetsov E A, Kagan M Yu, Turlapov A V Phys. Rev. A 101 (4) (2020)
  71. Smolyakov M N Chaos, Solitons & Fractals 132 109570 (2020)
  72. Zubarev N M, Kochurin E A Theor Math Phys 202 352 (2020)
  73. Ma D, Koval V, Jia Ch New J. Phys. 22 013046 (2020)
  74. Kuznetsov E A, Kagan M Yu Theor Math Phys 202 399 (2020)
  75. Levkov D  G, Panin A  G, Tkachev I  I Phys. Rev. D 102 (2) (2020)
  76. Chekhovskoy I S, Shtyrina O V et al Opt. Express 28 7817 (2020)
  77. Bulanov S  V, Sasorov P  V et al Phys. Rev. D 101 (1) (2020)
  78. D’Ambroise J, Kevrekidis P G Phys. Scr. 94 115203 (2019)
  79. Chekhovskoy I S, Sidelnikov O S et al Handbook of Optical Fibers Chapter 15 (2019) p. 317
  80. Konyukhov A I, Shchurkin E V et al J. Exp. Theor. Phys. 128 384 (2019)
  81. Djoko M, Kofane T C Communications In Nonlinear Science And Numerical Simulation 68 169 (2019)
  82. Sary G, Gremillet L, Canaud B 26 (7) (2019)
  83. Dingwall R J, Öhberg P Phys. Rev. A 99 (2) (2019)
  84. Alimenkov I V Theor Math Phys 201 1581 (2019)
  85. Goncharov V P 26 (9) (2019)
  86. Degasperis A, Lombardo S, Sommacal M Fluids 4 57 (2019)
  87. Cisneros-Ake L A, Carretero-González R et al Communications In Nonlinear Science And Numerical Simulation 74 268 (2019)
  88. Clarke S, Gorshkov K et al Physica D: Nonlinear Phenomena 366 43 (2018)
  89. Goncharov V P, Pavlov V I J. Exp. Theor. Phys. 126 276 (2018)
  90. Gao X, Zeng J Front. Phys. 13 (1) (2018)
  91. Vuillon L, Dutykh D, Fedele F Communications In Nonlinear Science And Numerical Simulation 57 202 (2018)
  92. Shtyrina O V, Kivshar Y S et al Advanced Photonics 2018 (BGPP, IPR, NP, NOMA, Sensors, Networks, SPPCom, SOF), (2018) p. JTu5A.45
  93. Kuznetsov E A Physics Letters A 382 2049 (2018)
  94. Selezov I T, Kryvonos Yu G, Gandzha I S Wave Propagation and Diffraction Foundations Of Engineering Mechanics Chapter 2 (2018) p. 25
  95. Abrashkin A A, Pelinovsky E N Uspekhi Fizicheskikh Nauk 188 329 (2018)
  96. [Abrashkin A A, Pelinovsky E N Phys.-Usp. 61 307 (2018)]
  97. Kartashov Ya V, Malomed B A et al Phys. Rev. A 98 (1) (2018)
  98. Shtyrina O V, Fedoruk M P et al Phys. Rev. A 97 (1) (2018)
  99. Chavanis P-H Phys. Rev. D 98 (2) (2018)
  100. Chekhovskoy I S, Sidelnikov O S et al Handbook of Optical Fibers Chapter 15-1 (2018) p. 1
  101. Kachulin D, Gelash A Nonlin. Processes Geophys. 25 553 (2018)
  102. Zuev L B Phys. Metals Metallogr. 118 810 (2017)
  103. Komarov F F Uspekhi Fizicheskikh Nauk 187 465 (2017)
  104. [Komarov F F Phys.-Usp. 60 435 (2017)]
  105. Levkov D  G, Panin A  G, Tkachev I  I Phys. Rev. Lett. 118 (1) (2017)
  106. Ablowitz M J, Ma Y-P, Rumanov I SIAM J. Appl. Math. 77 1248 (2017)
  107. Levkov D, Nugaev E, Popescu A J. High Energ. Phys. 2017 (12) (2017)
  108. Belashov V Yu, Belashova E S Geomagn. Aeron. 56 716 (2016)
  109. Chekhovskoy I S, Rubenchik A M et al Phys. Rev. A 94 (4) (2016)
  110. Lushchik A, Lushchik Ch et al Nuclear Instruments And Methods In Physics Research Section B: Beam Interactions With Materials And Atoms 374 90 (2016)
  111. Pushkarev A, Zakharov V Ocean Modelling 103 18 (2016)
  112. Pchelkina Y Z, Alimenkov I V J. Phys.: Conf. Ser. 738 012016 (2016)
  113. Zagorodnii A G, Kirichok A V, Kuklin V M Uspekhi Fizicheskikh Nauk 186 743 (2016)
  114. Nikitenkova S, Singh N, Stepanyants Y 25 (12) (2015)
  115. Goncharov V P, Pavlov V I Phys. Rev. E 91 (4) (2015)
  116. Goncharov V P, Pavlov V I Jetp Lett. 101 438 (2015)
  117. Sinkevich O A J. Exp. Theor. Phys. 121 321 (2015)
  118. Shablonin E, Popov A I et al Physica B: Condensed Matter 477 133 (2015)
  119. Zhang Y-Ch, Zhou Zh-W et al Phys. Rev. Lett. 115 (25) (2015)
  120. Zemlyanov A A, Bulygin A D, Geints Yu E Atmos Ocean Opt 27 463 (2014)
  121. Gandzha I S, Sedletsky Yu V, Dutykh D S Ukr. J. Phys. 59 1201 (2014)
  122. Borhanian J, Hosseini F F 21 (4) (2014)
  123. Zotov O D, Guglielmi A V, Sobisevich A L Izv., Phys. Solid Earth 49 882 (2013)
  124. POKLONSKI N A, VLASSOV A T et al Physics, Chemistry and Applications of Nanostructures, (2013) p. 36
  125. Kuznetsov E A, Passot T, Sulem P L Jetp Lett. 96 642 (2013)
  126. Goncharov V P, Pavlov V I J. Exp. Theor. Phys. 117 754 (2013)
  127. Bannikova E Yu, Kontorovich V M, Poslavsky S A J. Exp. Theor. Phys. 117 378 (2013)
  128. Postupaev V V, Burdakov A V et al 20 (9) (2013)
  129. Sazonov S V J. Exp. Theor. Phys. 117 885 (2013)
  130. Goncharov V P, Pavlov V I Phys. Rev. E 88 (2) (2013)
  131. Lushchik A, Lushchik Ch et al Physica Status Solidi (b) 250 261 (2013)
  132. Zaspa Yu P J. Frict. Wear 34 317 (2013)
  133. Sakbaev V Zh P-Adic Num Ultrametr Anal Appl 4 306 (2012)

© 1918–2025 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions