View source for Example Calculations for the KdV and IST

{{nonlinear waves course
 | chapter title = Example Calculations for the KdV and IST
 | next chapter = [[Reaction-Diffusion Systems]]
 | previous chapter = [[Connection betwen KdV and the Schrodinger Equation]]
}}

==Example1 Delta function potential.==

We have already calculated the scattering data for the delta function
potential. The scattering data is
<center><math>
S\left(  \lambda,0\right)  =\left(  k_{1},\sqrt{k_{1}},\frac{u_{0}}{2ik-u_{0}
},\frac{2ik}{2ik-u_{0}}\right)
</math></center>
The spectral data evolves as
<center><math>
k_{1}=k_{1}
</math></center>
<center><math>
c_{1}\left(  t\right)  =c_{1}\left(  0\right)  e^{4k_{1}^{3}t}=\sqrt{k_{1}
}e^{4k_{1}^{3}t}
</math></center>
<center><math>
r\left(  k,t\right)  =r\left(  k,0\right)  e^{8ik^{3}t}
</math></center>
<center><math>
a\left(  k,t\right)  =a\left(  k,0\right)
</math></center>
so that
<center><math>
S\left(  \lambda,t\right)  =\left(  k_{1},\sqrt{k_{1}}e^{4k_{1}^{3}t}
,\frac{u_{0}}{2ik-u_{0}}e^{8ik^{3}t},\frac{2ik}{2ik-u_{0}}\right)
</math></center>

==Example 2 A Hat Function==

We solve for the case when
<center><math>
u\left(  x\right)  =\left\{
\begin{matrix}
0, & x\notin\left[  -1,1\right]  \\
20, & x\in\left[  -1,1\right]
\end{matrix}
\right.
</math></center>
For the even solutions we need to solve
<center><math>
\tan\kappa=\frac{k}{\kappa}
</math></center>
where <math>\kappa=\sqrt{b-k^{2}}</math>.

For the odd solutions we need to solve and
<center><math>
\tan\kappa=-\frac{\kappa}{k}
</math></center>

Recall that the solitons have amplitude <math>2k_{n}^{2}</math> or <math>-2\lambda_{n}</math>. This
can be seen in the height of the solitary waves. 

We cannot work with a hat function numerically, because the jump in <math>u</math> leads
to high frequencies which dominate the response.. We can smooth our function
by a number of methods. We use here the function <math>\tanh\left(  x\right)  </math> so
we write
<center><math>
u\left(  x\right)  =\frac{20}{2}\left(  \tanh\left(  \nu\left(  x+1\right)
\right)  -\tanh\left(  \nu\left(  x+1\right)  \right)  \right)
</math></center>
where <math>\nu</math> is an appropriate constant to make the function increase in value
sufficiently rapidly but not too rapidly.

{| class="wikitable"
|-
! Animation
! Three-dimensional plot.
|-
| [[Image:Wide_function.gif|thumb|right|500px|Evolution of <math>u(x,t)</math>.]]
| [[Image:Widefunction.jpg|thumb|right|500px|Evolution of <math>u(x,t)</math> ]]

|}