The Korteweg-de Vries (KdV) equation is
a nonlinear 1+1D partial differential equation
that was originally derived to describe water waves.
It is usually given in its dimensionless form:
Where u(x,t) is the wave’s profile,
with x being the transverse coordinate.
The KdV equation notably has soliton solutions,
which can travel long distances without changing shape.
The derivation of the KdV equation starts in the same way as for
the Boussinesq wave equations;
the common parts will be discussed only briefly here.
Recall that Boussinesq set up two boundary conditions
at the liquid’s surface z=η(x,t).
Firstly, the kinematic boundary condition:
And secondly, the free surface boundary condition
from integrating the main Euler equation:
Where Ψ is the velocity potential u=∇Ψ,
with u=(u(x),u(z)) being 2D
due to the assumed symmetry along the y-axis.
Unlike Boussinesq, who assumed that p0=p at the surface,
de Vries decided to include surface tension using
the Young-Laplace law:
Where T is the energy cost per unit area,
and η is assumed to be slowly-varying such that ηx2
can be neglected in the curvature formula.
His free surface condition was thus:
Then, like Boussinesq, de Vries differentiated this with respect to x, yielding:
The goal is to reduce the number of terms even further,
and then to combine these equations into one.
To do this, the method of successive approximations is used:
first, a linearized version of the problem is solved,
which is easily shown to give Lagrange’s result:
Where η+ and η− are arbitrary functions
that respectively represent forward- and backward-propagating waves.
Then this result is used to derive a higher-order equation.
At this point, the calculations of Boussinesq and de Vries diverge.
Boussinesq kept using static Cartesian coordinates
and assumed a forward-moving wave η(x−ght),
whereas de Vries chose a reference frame moving at a speed q0.
However, the way de Vries did this is somewhat unusual:
rather than transform the coordinate system,
the velocity is incorporated into his ansatz for f;
in other words, he assumed that the entire liquid is moving at q0.
For a wave going in the positive x-direction,
the linearized problem then predicts a profile η(x−(gh+q0)),
so de Vries chose q0=−gh to make it stationary.
Analogously, q0=gh for a backward-moving wave.
With this in mind, the ansatz is:
Where α is a constant parameter
(which we will use to handle velocity discrepancies
between the linear and nonlinear theories).
The correction represented by γ is much smaller,
We insert this ansatz into the above equations, yielding:
We keep terms on the order of αη,
but neglect anything smaller (ηγ etc.),
because by assumption we have h≫η≫α≫γ.
Furthermore, each x-derivative is roughly equivalent to dividing by λ,
and since the water is shallow (λ≫h)
successive differentiations reduce terms’ magnitudes,
so terms like αη and η2 are kept
only if they contain at most one x-derivative:
e.g. ηηx stays, but ηx2 does not.
This reduces the equations to the following:
This leads to the original Korteweg-de Vries equation for waves on shallow water:
Where we have defined the dispersion parameter σ as follows:
What about α?
Looking at the ansatz for f, we see that
the body of water is already assumed to be moving at q0,
minus gα/q0, so by varying α
we are modifying the water’s velocity.
The term in the KdV equation simply corrects for our chosen value of α.
It has no deeper meaning than that: for any value of α,
the full range of KdV solutions can still be obtained.
Let us derive the standard non-dimensionalized form
of the KdV equation seen in most literature.
To do so, we make the following coordinate transformation,
where η~, x~ and t~ are dimensionless,
and ηc, xc, tc and vc are free dimensioned scale parameters:
The original derivatives with respect to x and t are then rewritten like so:
Now we must choose the scale parameters’ values.
By convention, the second term is removed,
the third has a factor 6, and the last has a factor −1,
This is pure convention; other choices are valid too.
Reducing these equations:
To proceed, we need to take the square root of xc2,
but we must make sure that xc2>0, because all quantities are real.
We enforce this in our choice of ηc, where s≡sgn(σ):
These are the final scale parameter values,
leading to the desired dimensionless form:
Recall that α sets the background fluid velocity,
and vc controls the coordinate system’s motion:
our choice of vc simply cancels out the effect of α.
This reveals the point of α:
the KdV equation has solutions moving at various speeds,
so, for a given η, we can always choose α (and hence vc)
such that the wave appears stationary.
Let us make the following ansatz for the dimensionless wave profile η~,
assuming there exists a solution that maintains its shape
while propagating at a constant “velocity” v:
Inserting this into the dimensionless KdV equation
tells us that ϕ must satisfy:
Integrating this equation and introducing an integration constant A/2 gives:
Let us restrict our search further by demanding
that ϕ→0 and ϕξ→0 for ξ→±∞.
Clearly, that implies ϕξξ→0, so we must set A=0.
We will do so shortly; first multiply by ϕξ:
Cleaning up and isolating for η gives the form below.
Remember that v is dimensionless:
We are almost finished, and could leave the solution in this form if we wanted to.
However, this function contains two free parameters, v and α,
and it would be nice to combine them into one
(which is indeed possible without losing information).
From looking at the expression, it is clear that both v and α
control how fast the soliton moves in our reference frame.
As discussed earlier, α simply modifies the bulk fluid velocity,
so could we relate v and α such that the soliton appears stationary?
Yes, by demanding:
Recall that v>0 to get a stable dimensionless solution:
this result therefore tells us that α and σ should have opposite signs.
That requirement is actually equivalent to v>0, and can be found directly
by deriving the ϕξ2=P(ϕ) equation without non-dimensionalization.
At last, this brings us to the general stationary soliton, given by:
For σ>0 and α<0 the amplitude is positive;
the wave is a “bump” on the water, as you would expect.
However, for σ<0 and α>0 the amplitude is negative,
so then the wave is actually a “dip”, which may be surprising.
For water, the condition σ<0 equates to h≲0.5cm,
so such waves are indeed hard to observe.