KdV_Chapter_final.pdf (305.11 kB)

# Korteweg - de Vries equation: solitons and undular bores

chapter

posted on 2006-01-27, 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.

## History

## School

- Science

## Department

- Mathematical Sciences

## Pages

292856 bytes## Citation

EL, G.A., 2007. Korteweg - de Vries equation: solitons and undular bores [Chapter 2]. IN: Grimshaw, R.J.H. (ed.). Solitary Waves in Fluids: Advances in Fluid Mechanics. Southampton: WIT Press, pp.19-53.## Publication date

2007## Notes

This version of the book chapter has been made available, courtesy of WIT Press, from the book IN: Grimshaw, R.J.H. (ed.). Solitary Waves in Fluids: Advances in Fluid Mechanics. Southampton: WIT Press, pp.19-53.## ISBN

1845641574## Book series

WIT Transactions on State of the Art in Science and Engineering; 9## Language

- en