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Download file# Korteweg - de Vries equation: solitons and undular bores

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posted on 27.01.2006, 11:31 authored by Gennady ElThe Korteweg – de Vries (KdV) equation is a fundamental mathematical
model for the description of weakly nonlinear long wave propagation
in dispersive media. It is known to possess a number of families of exact
analytic solutions. Two of them: solitons and nonlinear periodic travelling
waves – are of particular interest from the viewpoint of fluid dynamics
applications as they occur as typical asymptotic outcomes in a broad class
of initial/boundary-value problems. Two different major approaches have
been developed in the last four decades to deal with the problems involving
solitons and nonlinear periodic waves: inverse scattering transform and
the Whitham method of slow modulations. We review these methods and
show relations between them. Emphasis is made on solving the KdV equation
with large-scale initial data. In this case, the long-time evolution of
an initial perturbation leads to formation of an expanding undular bore,
a modulated travelling wave connecting two different non-oscillating flows.
Another problem considered is the propagation of a soliton through a variable
environment in the framework of the variable-coefficient KdV equation.
If the background environment varies slowly, the solitary wave deforms adiabatically
and an extended small-amplitude trailing shelf is generated. On
a long-time scale, the trailing shelf evolves, via an intermediate stage of an
undular bore, into a secondary soliton train.

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- Mathematical Sciences