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Nonlinear Coherent Phenomena in
Continuous Media
. sov
. Zakharov
ABSTRACT This review is devoted to description of coherent nonlin-
ear phenomena in almost conservative media with applications to plasma
physics, fluid dynamics and nonlinear optics. The main attention in the
review is paid to consideration of solitons, collapses, and black holes. The
latter is a quasi-stationary singular object which appear after the forma-
tion of a singularity in nonlinear wave systems. We discuss in details the
qualitative reasons of the wave collapse and a difference between solitons
and collapses, and apply to their analysis exact methods based on the
integral estimates and the Hamiltonian formalism. These approaches are
demonstrated mainly on the basic nonlinear models, . on the nonlinear
Schr¨odinger equation and the Kadomtsev-Petviashvili equation and their
generalizations.
1 Introduction
All real continuous media, including vacuum, are nonlinear. Nonlinearity
might be a cause of quite opposite physical effects. One of them is phase
randomization leading to formation of a chaotic state - weak or strong
wave turbulence. Wind-driven waves on the ocean surface is the classical
example of that sort. Another group of effects is spontaneous generation
of coherent structures. These structures may be localized in space or both
in space and in time. Phases of Fourier harmonics, forming the structures,
are strongly correlated.
Very often coherent structures coexist with wave turbulence. A simple
example of the coherent structure is ‘white caps’ on the crest of gravity
wave of high amplitude. Elementary visual observation shows that just
before breaking, a wave crest takes the universal, wedge-type shape. Ap-
parently, the posing this shape have correlated phases. The
wave breaking is an important mechanism of energy and momentum dis-
sipation on the ocean. A satisfactory theory of this basic effe