Rational solution to Korteweg–De Vries KdV equation

Students seeing differential equations for the first time expect every equation to have a nice closed-form solution, because up to that point in their education nearly every problem they’ve seen has been contrived to have a nice closed-form solution.

Once you resign yourself to the fact that a differential equation will rarely have a closed form solution, it’s a treat when you run across one that does. This is especially true for nonlinear equations.

The Korteweg–De Vries (KdV) equation is

$u_t - 6 u\, u_x + u_{xxx} = 0$

is such a treat. I wrote a few days ago about the sech² solution to the KdV equation.

$u(x,t) = -\frac{v}{2} \,\text{sech}^2\left(\frac{\sqrt{v}}{2} (x - vt - a)\right )$

There’s also a rational solution:

$u(x, t) = 6 x \frac{ \left(x^3-24 t\right)}{\left(x^3 + 12 t\right)^2}$

We can verify this is a solution to the KdV equation reusing the Mathematica code from the earlier post.

    u[x_, t_] := u[x_, t_] := 6 x (x^3 - 24 t)/(x^3 + 12 t)^2
    Simplify[ D[u[x, t], {t, 1}] 
            - 6 u[x, t] D[u[x, t], {x, 1}] 
            + D[u[x, t], {x, 3}] ]

This simplifies to 0.

Here’s a plot:

The top of the plot looks like a two-lane road on top of a mountain ridge, with a sinkhole in the middle of the road.

The “road” is a artifact of plotting. The solution is singular along the curve x³ + 12t= 0, and Mathematica had to chop the top of the graph off because it can’t plot an infinitely tall function.