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. Pirogova2, 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
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

:   ,    «  : » 182 569–592 (2012); DOI: 10.3367/UFNr.0182.201206a.0569

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

  1. S.A. Akhmanov, A.P. Sukhorukov, R.V. Khokhlov “Self-focusing and diffraction of light in a Nonlinear medium10 609–636 (1968)
  2. S.K. Turitsyn, N.N. Rozanov et alDissipative solitons in fiber lasers59 642–668 (2016)
  3. S.A. Akhmanov, V.A. Vysloukh, A.S. Chirkin “Self-action of wave packets in a nonlinear medium and femtosecond laser pulse generation29 642–647 (1986)
  4. M.Yu. Kagan, A.V. Turlapov “BCS—BEC crossover, collective excitations, and hydrodynamics of superfluid quantum fluids and gases62 215–248 (2019)
  5. V.G. Makhan’kov, Yu.P. Rybakov, V.I. Sanyuk “Localised nontopological structures: construction of solutions and stability problems37 113–137 (1994)
  6. S.V. Chekalin, V.P. Kandidov “From self-focusing light beams to femtosecond laser pulse filamentation56 123–140 (2013)
  7. N.A. Veretenov, N.N. Rosanov, S.V. Fedorov “Laser solitons: topological and quantum phenomena65 131–162 (2022)
  8. P.K. Shukla, B. Eliasson “Nonlinear aspects of quantum plasma physics53 51–76 (2010)
  9. A.I. Zhakin “Electrohydrodynamics of charged surfaces56 141–163 (2013)
  10. F.F. Komarov “Defect and track formation in solids irradiated by superhigh-energy ions46 1253–1282 (2003)
  11. T.I. Belova, A.E. Kudryavtsev “Solitons and their interactions in classical field theory40 359–386 (1997)
  12. A.M. Miterev “Theoretical aspects of the formation and evolution of charged particle tracks45 1019–1050 (2002)
  13. V.E. Zakharov, E.A. Kuznetsov “Hamiltonian formalism for nonlinear waves40 1087–1116 (1997)
  14. V.A. Aleshkevich, G.D. Kozhoridze, A.N. Matveev “Self-action of partly coherent laser radiation34 (9) 777–803 (1991)
  15. B.S. Kerner, V.V. Osipov “Autosolitons32 101–138 (1989)
  16. M.V. Nezlin “Rossby solitons (Experimental investigations and laboratory model of natural vortices of the Jovian Great Red Spot type)29 807–842 (1986)
  17. S.A. Akhmanov “Khokhlov’s method in the theory of nonlinear waves29 589–606 (1986)
  18. G.I. Kanel, V.E. Fortov, S.V. Razorenov “Shock waves in condensed-state physics50 771–791 (2007)
  19. A.V. Guglielmi “Ultra-low-frequency electromagnetic waves in the Earth’s crust and magnetosphere50 1197–1216 (2007)
  20. V.F. Kop’ev, S.A. Chernyshev “Vortex ring oscillations, the development of turbulence in vortex rings and generation of sound43 663–690 (2000)

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