文档介绍：This is page 47
Printer: Opaque this
2
Perturbation Theories for Nonlinear
Waves
Lev Ostrovsky
Konstantin Gorshkov
ABSTRACT Some ideas and theories developed since 1960s to describe
nonlinear waves with slowly varying parameters (modulated waves)are out-
lined. These theories are associated with different versions of the asymp-
totic perturbation method. In this framework, both quasi-periodic and soli-
tary waves (solitons)can be treated. A scheme for reduction of a quasi-
hyperbolic system to one or more evolution equations is also presented.
Some challenges for the theory are briefly discussed.
1 Introduction
Among the most remarkable achievements of nonlinear wave theory by
the eve of the 21st century,many of us will cite the development of exact
methods such as the inverse scattering method in the theory of integrable
nonlinear wave equations. At the same time,among the most effective tools
for solving practical problems (apart from direct numerical simulations),
the outstanding role of approximate analytical methods will probably be
appreciated as well.
Perturbation methods in mechanics had been developed in the 19th cen-
tury for astronomical applications. Later,they occupied an outstanding
place in quantum physics,to mention only two very important applications.
These methods were first elaborated for the cases when small perturbations
in the equations or initial conditions would result in small changes in the
solutions. However,in many cases,small perturbations may change the so-
lutions rather strongly,and the smallness of the former is reflected only in
the slowness of the latter’s pared to the characteristic time
scale of the process. An adequate tool for solving such problems is the
asymptotic perturbation theory,which constructs a series in a small pa-
rameter that may not converge,but its lower-order terms turn out to be
the closer to the exact solution,the smaller is the expansion parameter.
The main term of such