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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.

Text can be downloaded in Russian. English translation is available on IOP Science.
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/
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

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

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