Linear KdV dispersion

The Korteweg–De Vries (KdV) equation

is a nonlinear PDE used to model shallow water waves. The linear counterpart omits the nonlinear term in the middle.

This variant is useful in itself, but also for understanding the nonlinear KdV equation.

Solitons

Solutions to the linear KdV equation spread out over time. The nonlinear term in the KdV equation counterbalances the dispersion term u_xxx so that solutions to the nonlinear PDE behave in some ways like linear solutions.

Solutions to a nonlinear equation don’t add, and yet they sorta add. Here’s a description from [1].

At the heart of these observations is the discovery that these nonlinear waves can interact strongly and then continue thereafter almost as if there had been no interaction at all. This persistence of the wave led Zabusky and Kruskal to coin the term ‘soliton’ (after photon, proton, etc.) to emphasize the particle-like character of these waves which retain their identities in a collision.

I added the emphasis on almost because many descriptions leave out this important qualifier.

Solution to linear KdV

There is a compact expression for the solution to the linear KdP equation if u, u_x , and u_xx all go to 0 as |x| → ∞. If u(x, 0) = f(x), then the solution is

Here Ai(z) is the Airy function. This function has come up several times here. For example, I’ve written about the Airy function in the context of triple factorials and in connection with Bessel functions.

Aside on third order equations

Third order differential equations are uncommon. Third order linear ODEs are quite rare. Third order differential equations are usually nonlinear PDEs, like the KdV equation. The linear KdV is an example of a linear third order PDE that arises in applications.

[1] P. G. Drazin and R. S. Johnson. Solitons: an introduction. Cambridge University Press. 1989.