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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/
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)
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Received: 14th, July 2011, 2nd, August 2011

Îðèãèíàë: Çàõàðîâ Â Å, Êóçíåöîâ Å À «Ñîëèòîíû è êîëëàïñû: äâà ñöåíàðèÿ ýâîëþöèè íåëèíåéíûõ âîëíîâûõ ñèñòåì» ÓÔÍ 182 569–592 (2012); DOI: 10.3367/UFNr.0182.201206a.0569

References (103) Cited by (142) ↓ Similar articles (20)

  1. Dong L, Fan M et al Chaos, Solitons & Fractals 202 117618 (2026)
  2. Ruban V P Jetp Lett. 121 (11) 824 (2025)
  3. Morris S J, Ho Ch J et al Phys. Rev. A 111 (4) (2025)
  4. Sun Y, Parra-Rivas P et al Phys. Rev. Research 7 (3) (2025)
  5. Lashkin V M, Cheremnykh O K Physics of Plasmas 32 (2) (2025)
  6. Didenkulova E, Flamarion M V, Pelinovsky E Physica D: Nonlinear Phenomena 481 134815 (2025)
  7. Yao X, Wang L Nonlinear Dyn 113 (10) 11907 (2025)
  8. Flamarion M V, Pelinovsky E, Didenkulova E Phys. Wave Phen. 33 (1) 9 (2025)
  9. Geints Yu E, Bulygin A D et al Phys. Rev. A 112 (4) (2025)
  10. Chen Sh, Wang L et al Chaos, Solitons & Fractals 190 115777 (2025)
  11. AL-Taie Mohammed Salim Jasim J Comput Electron 24 (4) (2025)
  12. Muddassar M, Khan A G et al Bound Value Probl 2025 (1) (2025)
  13. Ruban V P Jetp Lett. 121 (5) 354 (2025)
  14. Turco E, Bilotta A Continuum Mech. Thermodyn. 37 (4) (2025)
  15. Vitanov N K, Vitanov K N Symmetry 17 (8) 1363 (2025)
  16. 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
  17. Yao X, Ma J, Meng G Nonlinear Dyn 112 (20) 18435 (2024)
  18. Ruban V P Pisʹma V žurnal êksperimentalʹnoj I Teoretičeskoj Fiziki 119 (7-8) 579 (2024)
  19. Kochurin E  A, Kuznetsov E  A Phys. Rev. Lett. 133 (20) (2024)
  20. Jin X-W, Yang Zh-Y et al Phys. Rev. B 109 (13) (2024)
  21. 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
  22. Malomed B A Advances In Physics: X 9 (1) (2024)
  23. Ruban V P Jetp Lett. 119 (8) 585 (2024)
  24. Lashkin V M Phys. Rev. E 109 (6) (2024)
  25. Zhong M, Chen Y et al Proc. R. Soc. A. 480 (2282) (2024)
  26. Gelash A, Dremov S et al Phys. Rev. Lett. 132 (13) (2024)
  27. Ruban V P Jetp Lett. 120 (10) 713 (2024)
  28. Lashkin V M Physics of Plasmas 31 (4) (2024)
  29. Jin X-W, Yang Zh-Y et al Phys. Rev. B 109 (1) (2024)
  30. Ostrovsky L, Pelinovsky E et al Chaos: An Interdisciplinary Journal of Nonlinear Science 34 (6) (2024)
  31. Elkamash I S, Reville B et al Chaos, Solitons & Fractals 188 115531 (2024)
  32. Lashkin V M, Cheremnykh O K Physics of Fluids 36 (2) (2024)
  33. Dong L, Fan M, Malomed B A Chaos, Solitons & Fractals 188 115499 (2024)
  34. Ruban V P Pisʹma V žurnal êksperimentalʹnoj I Teoretičeskoj Fiziki 120 (9-10) 745 (2024)
  35. Lashkin V M, Cheremnykh O K Phys. Rev. E 110 (2) (2024)
  36. Liu D, Gao Ya et al Optics & Laser Technology 177 111181 (2024)
  37. Zemlyanov A A, Minina O V Atmos Ocean Opt 36 (4) 314 (2023)
  38. Lashkin V M, Cheremnykh O K et al Phys. Rev. E 107 (2) (2023)
  39. Kukushkin A B, Kulichenko A A Foundations 3 (3) 602 (2023)
  40. Dong L, Fan M Chaos, Solitons & Fractals 173 113728 (2023)
  41. Kumar Sh, Li P, Malomed B A Phys. Rev. E 108 (2) (2023)
  42. Levkov D  G, Maslov V  E Phys. Rev. D 108 (6) (2023)
  43. Kuznetsov E A Radiophys Quantum El 66 (5-6) 305 (2023)
  44. Kagan M Yu, Aksenov S V et al Jetp Lett. 117 (10) 755 (2023)
  45. Tribelsky M I Proc. R. Soc. A. 479 (2277) (2023)
  46. Zemlyanov A A, Minina O V et al XVI International Conference on Pulsed Lasers and Laser Applications, (2023) p. 16
  47. Li Ch, Konotop V V et al Chaos, Solitons & Fractals 174 113848 (2023)
  48. Kagan M Yu, Aksenov S V et al Pisʹma V žurnal êksperimentalʹnoj I Teoretičeskoj Fiziki 117 (9-10 (5)) 754 (2023)
  49. Chen Zh, Li Y et al Communications In Nonlinear Science And Numerical Simulation 118 107013 (2023)
  50. Kontorovich V M, Poslavskyi S A Low Temperature Physics 48 (5) 413 (2022)
  51. Kuznetsov E A J. Exp. Theor. Phys. 135 (1) 121 (2022)
  52. Malomed B A Multidimensional Solitons (2022) p. 9-1
  53. Chen Zh, Li Y et al SSRN Journal (2022)
  54. Deng D, Yuan Zh et al Geophysical Research Letters 49 (4) (2022)
  55. Malomed B A Multidimensional Solitons (2022) p. 1-1
  56. Malomed B A Low Temperature Physics 48 (11) 856 (2022)
  57. Mezentsev V K, Podivilov E et al Phys. Rev. E 106 (5) (2022)
  58. Fokas A S, Cao Yu, He J Fractal Fract 6 (8) 425 (2022)
  59. Rao J, He J, Malomed B A Journal of Mathematical Physics 63 (1) (2022)
  60. Agafontsev D S, Kuznetsov E A et al Phys.-Usp. 65 (2) 189 (2022)
  61. Levkov D G, Maslov V E et al J. High Energ. Phys. 2022 (12) (2022)
  62. Garani R, Levkov D, Tinyakov P Phys. Rev. D 105 (6) (2022)
  63. Vitanov N K Entropy 24 (11) 1653 (2022)
  64. Bica I, Mucalica A Analele Universitatii "Ovidius" Constanta - Seria Matematica 30 (2) 45 (2022)
  65. Belashov V Yu, Kharshiladze O A, Belashova E S Geomagn. Aeron. 61 (2) 149 (2021)
  66. Sinkevich O A High Temp 59 (1) 77 (2021)
  67. Kochurin E A, Zubarev N M Fluids 6 (3) 125 (2021)
  68. Khalili S, Hasanbeigi A, Sobhanian S Plasma Phys. Rep. 47 (3) 298 (2021)
  69. PELAP François Beceau, NDECFO Jean Emac, DEFFO Guy Roger Phys. Scr. 96 (7) 075211 (2021)
  70. Zuev L B Multiscale Biomechanics and Tribology of Inorganic and Organic Systems Springer Tracts In Mechanical Engineering Chapter 12 (2021) p. 245
  71. Guo L, Chabchoub A, He J Physica D: Nonlinear Phenomena 426 132990 (2021)
  72. Fonkoua S A T, Pelap F B et al Eur. Phys. J. Plus 136 (4) (2021)
  73. Kuznetsov E A, Kagan M Yu J. Exp. Theor. Phys. 132 (4) 704 (2021)
  74. Mullyadzhanov R I, Gelash A A Radiophys Quantum El 63 (9-10) 786 (2021)
  75. Dmitriev A  S, Levkov D  G et al Phys. Rev. D 104 (2) (2021)
  76. Rao J, Chow K W et al Stud Appl Math 147 (3) 1007 (2021)
  77. Kuznetsov E A, Kagan M Yu Theor Math Phys 202 (3) 399 (2020)
  78. Zubarev N M, Kochurin E A Theor Math Phys 202 (3) 352 (2020)
  79. Smolyakov M N Chaos, Solitons & Fractals 132 109570 (2020)
  80. Chekhovskoy I S, Shtyrina O V et al Opt. Express 28 (6) 7817 (2020)
  81. Bulanov S  V, Sasorov P  V et al Phys. Rev. D 101 (1) (2020)
  82. Nugaev E Ya, Shkerin A V J. Exp. Theor. Phys. 130 (2) 301 (2020)
  83. Kuznetsov E A, Kagan M Yu, Turlapov A V Phys. Rev. A 101 (4) (2020)
  84. Chavanis P-H Phys. Rev. D 102 (8) (2020)
  85. Oloo J O, Shrira V I Teoreticheskaya Matematicheskaya Fizika 203 (1) 91 (2020) [Oloo J O, Shrira V I Theor Math Phys 203 (1) 512 (2020)]
  86. Ma D, Koval V, Jia Ch New J. Phys. 22 (1) 013046 (2020)
  87. Levkov D  G, Panin A  G, Tkachev I  I Phys. Rev. D 102 (2) (2020)
  88. Alfimov G L, Fedotov A P, Sinelshchikov D I Physica D: Nonlinear Phenomena 402 132245 (2020)
  89. Dingwall R J, Öhberg P Phys. Rev. A 99 (2) (2019)
  90. Chekhovskoy I S, Sidelnikov O S et al Handbook of Optical Fibers Chapter 15 (2019) p. 317
  91. Degasperis A, Lombardo S, Sommacal M Fluids 4 (1) 57 (2019)
  92. Djoko M, Kofane T C Communications In Nonlinear Science And Numerical Simulation 68 169 (2019)
  93. Konyukhov A I, Shchurkin E V et al J. Exp. Theor. Phys. 128 (3) 384 (2019)
  94. Alimenkov I V Theor Math Phys 201 (2) 1581 (2019)
  95. D’Ambroise J, Kevrekidis P G Phys. Scr. 94 (11) 115203 (2019)
  96. Cisneros-Ake L A, Carretero-González R et al Communications In Nonlinear Science And Numerical Simulation 74 268 (2019)
  97. Sary G, Gremillet L, Canaud B Physics of Plasmas 26 (7) (2019)
  98. Goncharov V P Physics of Plasmas 26 (9) (2019)
  99. Vuillon L, Dutykh D, Fedele F Communications In Nonlinear Science And Numerical Simulation 57 202 (2018)
  100. Shtyrina O V, Fedoruk M P et al Phys. Rev. A 97 (1) (2018)
  101. Gao X, Zeng J Front. Phys. 13 (1) (2018)
  102. Selezov I T, Kryvonos Yu G, Gandzha I S Wave Propagation and Diffraction Foundations Of Engineering Mechanics Chapter 2 (2018) p. 25
  103. Shtyrina O V, Kivshar Y S et al Advanced Photonics 2018 (BGPP, IPR, NP, NOMA, Sensors, Networks, SPPCom, SOF), (2018) p. JTu5A.45
  104. Kuznetsov E A Physics Letters A 382 (31) 2049 (2018)
  105. Abrashkin A A, Pelinovsky E N Uspekhi Fizicheskikh Nauk 188 (03) 329 (2018) [Abrashkin A A, Pelinovsky E N Phys.-Usp. 61 (3) 307 (2018)]
  106. Chavanis P-H Phys. Rev. D 98 (2) (2018)
  107. Kachulin D, Gelash A Nonlin. Processes Geophys. 25 (3) 553 (2018)
  108. Goncharov V P, Pavlov V I J. Exp. Theor. Phys. 126 (2) 276 (2018)
  109. Clarke S, Gorshkov K et al Physica D: Nonlinear Phenomena 366 43 (2018)
  110. Kartashov Ya V, Malomed B A et al Phys. Rev. A 98 (1) (2018)
  111. Chekhovskoy I S, Sidelnikov O S et al Handbook of Optical Fibers Chapter 15-1 (2018) p. 1
  112. Zuev L B Phys. Metals Metallogr. 118 (8) 810 (2017)
  113. Komarov F F Uspekhi Fizicheskikh Nauk 187 (05) 465 (2017) [Komarov F F Phys.-Usp. 60 (5) 435 (2017)]
  114. Ablowitz M J, Ma Y-P, Rumanov I SIAM J. Appl. Math. 77 (4) 1248 (2017)
  115. Levkov D  G, Panin A  G, Tkachev I  I Phys. Rev. Lett. 118 (1) (2017)
  116. Levkov D, Nugaev E, Popescu A J. High Energ. Phys. 2017 (12) (2017)
  117. Chekhovskoy I S, Rubenchik A M et al Phys. Rev. A 94 (4) (2016)
  118. Belashov V Yu, Belashova E S Geomagn. Aeron. 56 (6) 716 (2016)
  119. Pchelkina Y Z, Alimenkov I V J. Phys.: Conf. Ser. 738 012016 (2016)
  120. Lushchik A, Lushchik Ch et al Nuclear Instruments And Methods In Physics Research Section B: Beam Interactions With Materials And Atoms 374 90 (2016)
  121. Pushkarev A, Zakharov V Ocean Modelling 103 18 (2016)
  122. Zagorodnii A G, Kirichok A V, Kuklin V M Uspekhi Fizicheskikh Nauk 186 (7) 743 (2016)
  123. Goncharov V P, Pavlov V I Jetp Lett. 101 (7) 438 (2015)
  124. Sinkevich O A J. Exp. Theor. Phys. 121 (2) 321 (2015)
  125. Zhang Y-Ch, Zhou Zh-W et al Phys. Rev. Lett. 115 (25) (2015)
  126. Nikitenkova S, Singh N, Stepanyants Y Chaos: An Interdisciplinary Journal of Nonlinear Science 25 (12) (2015)
  127. Goncharov V P, Pavlov V I Phys. Rev. E 91 (4) (2015)
  128. Shablonin E, Popov A I et al Physica B: Condensed Matter 477 133 (2015)
  129. Zemlyanov A A, Bulygin A D, Geints Yu E Atmos Ocean Opt 27 (6) 463 (2014)
  130. Borhanian J, Hosseini F F Physics of Plasmas 21 (4) (2014)
  131. Gandzha I S, Sedletsky Yu V, Dutykh D S Ukr. J. Phys. 59 (12) 1201 (2014)
  132. Kuznetsov E A, Passot T, Sulem P L Jetp Lett. 96 (10) 642 (2013)
  133. Zaspa Yu P J. Frict. Wear 34 (4) 317 (2013)
  134. POKLONSKI N A, VLASSOV A T et al Physics, Chemistry and Applications of Nanostructures, (2013) p. 36
  135. Goncharov V P, Pavlov V I Phys. Rev. E 88 (2) (2013)
  136. Lushchik A, Lushchik Ch et al Physica Status Solidi (b) 250 (2) 261 (2013)
  137. Postupaev V V, Burdakov A V et al Physics of Plasmas 20 (9) (2013)
  138. Bannikova E Yu, Kontorovich V M, Poslavsky S A J. Exp. Theor. Phys. 117 (2) 378 (2013)
  139. Zotov O D, Guglielmi A V, Sobisevich A L Izv., Phys. Solid Earth 49 (6) 882 (2013)
  140. Goncharov V P, Pavlov V I J. Exp. Theor. Phys. 117 (4) 754 (2013)
  141. Sazonov S V J. Exp. Theor. Phys. 117 (5) 885 (2013)
  142. Sakbaev V Zh P-Adic Num Ultrametr Anal Appl 4 (4) 306 (2012)
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