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1.3.4 Wavebreaking
Slide : [
wavebreaking 
shockwaves 
solitons 
VIDEO
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Wavebreaking is a nonlinearity that is particularly nicely
understood when surfing on a see shore, where shallow waters steepen
the waves until they break. The process can be modeled theoretically
from the advection equation by choosing the
velocity proportional to the amplitude
:

(1.3.4#eq.1) 
Since a local maximum (large
) propagates faster than a local minimum
(small
), the top of the wave tries to overtake the bottom.
The function soon becomes multivalued causing the wave (and our numerical
schemes) to break.
Sometimes, the wavebreaking is balanced by a competing mechanism.
This is the case of example in the Burger equation for
shockwaves

(1.3.4#eq.2) 
where the creation of a shock front (with short wavelengths) is physically
limited by diffusion, which damps the short wavelengths (1.3.2#eq.4).
Here is an example of
a shock formation computed using a 2levels explicit finite difference
scheme from chapter 2.
JBONE applet: press Start/Stop
to simulate the propagation of a shock front using the Burger equation,
where the wavebreaking nonlinearity is balanced by a finite diffusion.

Another type of nonlinear equations where the wavebreaking is balanced by
dispersion leads to the KortewegDeVries equation for solitons

(1.3.4#eq.3) 
The evolution below
shows how large amplitudes solitons (short wavelengths) propagate
faster than lower amplitudes (long wavelength), in agreement with the
dispersion analysis previously performed in
sect.1.3.3.
JBONE applet: press Start/Stop
to simulate the propagation of solitons using the KdV equation,
where the wavebreaking nonlinearity is balanced by a finite
dispersion.

SYLLABUS Previous: 1.3.3 Dispersion
Up: 1.3 Prototype problems
Next: 1.3.5 Schrödinger
