KdV Cnoidal Wave Solutions
We will find a solution of the KdV equation for Shallow water waves,
The KdV equation has two qualitatively different types of permanent form travelling wave solution.
These are referred to as cnoidal waves and solitary waves.
KdV equation in space
Assume we have wave travelling with speed without change of form,
and substitute into KdV equation then we obtain
where is the travelling wave coordinate.
We rearrange and integrate this equation with respect to to give
then multiply to all terms and integrate again
Standardization of KdV equation
We define , so
It turns out that we require 3 real roots to obtain periodic solutions. Let roots be .
We can imagine the graph of cubic function which has 3 real roots and we can now write a function
From the equation , we require
We are only interested in solution for and we need .
and now solve equation in terms of the roots
We define , and obtain
crest to be at and
and a further variable Y via
which is separable.
In order to get this into a completely standard form we define
Solution of the KdV equation
A simple quadrature of equation (1) subject to the condition (2) the gives us
Jacobi elliptic function can be written in the form
Now we can write Y with fixed values of , as
is another Jacobi elliptic function with , and waves are called "cnoidal waves".
Using the result , our final result can be expressed in the form