Solitons and collapses: Two evolution scenarios of nonlinear wave systems

V. E. Zakharov, E. A. Kuznetsov

    Research output: Contribution to journalReview articlepeer-review

    84 Citations (Scopus)

    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 languageEnglish
    Pages (from-to)535-556
    Number of pages22
    JournalPhysics-Uspekhi
    Volume55
    Issue number6
    DOIs
    Publication statusPublished - 1 Jan 2012

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