KdV Equation Derivation
We consider the method of derivation of KdV Equation in the concept of Nonlinear Shallow Water Waves.
In the analysis of Nonlinear Shallow Water Waves equations we see that there are two important geometrical parameters, and involved. By choosing appropriate magnitudes for and , we can consider a theory in which dispersion and nonlinearity are in balance. The Korteweg-de Vries Equation verifies the relation between dispersion and nonlinearity properties.
We begin with the equations for waves on water,
Provided that at we have,
To make these equations dimensionless, we use the scaled variables,
where is defined as linear wave speed in shallow water. Hence the dimensionless system is,
where and are two small parameters which are given in this problem.
In the next step we use the transform and introduce further transformation to remove from the equations,
The key idea is that is .
The boundary condition (4) expresses at the flat bed, . The boundary condition (3) is Bernoulli equation and (2) is kinematic boundary condition. Now we use asymptotic expansions of the form,
to derive an equation for each according to the boundary conditions (2) to (4).
* Derivation of 's:
Substituting (5) and (6), (1) must be true for all powers of . Therefore,
Now at leading order the Bernoulli and kinematic equations, (3) and (2), gives,
Differentiating (b) and eliminating and from (c) allow us to write,
Finally, (a) gives in terms of and hence
which is named Korteweg-de Vries (KdV) equation.
KdV equation includes dispersive effects through the term and nonlinear effects through the term and governs the behavior of the small amplitude waves, with . It is reasonable to ask when and where the independent variables, and , are of in order to determine more precisely the region in physical space where the KdV equation is valid as an approximation of the actual flow. According to the definition of and , if , then and . This leads us to interpret any waveform that arises as a solution of the KdV equation as the large time limit of an initial value problem.
For solution of KdV equation please refer here.