### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 535-556 |

Number of pages | 22 |

Journal | Physics-Uspekhi |

Volume | 55 |

Issue number | 6 |

DOIs | |

State | Published - 2012 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Physics-Uspekhi*,

*55*(6), 535-556. https://doi.org/10.3367/UFNr.0182.201206a.0569

**Solitons and collapses : Two evolution scenarios of nonlinear wave systems.** / Zakharov, Vladimir E; Kuznetsov, E. A.

Research output: Contribution to journal › Article

*Physics-Uspekhi*, vol. 55, no. 6, pp. 535-556. https://doi.org/10.3367/UFNr.0182.201206a.0569

}

TY - JOUR

T1 - Solitons and collapses

T2 - Two evolution scenarios of nonlinear wave systems

AU - Zakharov, Vladimir E

AU - Kuznetsov, E. A.

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84866541257&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84866541257&partnerID=8YFLogxK

U2 - 10.3367/UFNr.0182.201206a.0569

DO - 10.3367/UFNr.0182.201206a.0569

M3 - Article

VL - 55

SP - 535

EP - 556

JO - Physics-Uspekhi

JF - Physics-Uspekhi

SN - 1063-7869

IS - 6

ER -