Generalized KdV Interactions

What happens when nonlinear waves interact?

In linear interactions of localized travelling waves, the initial wave profiles get superimposed as the waves collide and reappear as the waves move apart. Does the GKdV equation exhibit similar behavior for colliding waves?
This plot depicts the linear superposition of two traveling waves

The KdV Equation u_t + uu_x + u_xxx = 0

Among all of the GKdV-p equations u_t + u^pu_x + u_xxx = 0, the weakest nonlinearity occurs for the case p = 1, which is the KdV equation.

KdV travelling waves have the feature that their shape depends on their speed. Faster waves are taller and narrower, while shorter waves are shorter and broader. The ratio of height to width for a KdV travelling wave with speed c > 0 is approximately 3c : 2 / √c.

Travelling wave solutions of the KdV equation (p = 1)

This plot depicts the overlay of two travelling waves with speeds c₁ > c₂.

Collisions of KdV travelling waves are described by solutions whose initial profiles are superpositions of two (or more) individually travelling waves with different speeds. In these solutions, the waves undergo a nonlinear interaction that differs significantly compared to a linear superposition.

There are two kinds of collision behaviour, depending on the ratio of the initial wave speeds where c₁ and c₂ are the respective speeds of the faster and slower waves. When the speed ratio is c₁ / c₂ > 3, the waves first merge and then split apart into their previous shape and speed. In contrast, when the speed ratio is c₁ / c₂ < 3, the waves bounce and exchange both their shape and speed.

KdV (p = 1) colliding wave solution with speed ratio c₁ / c₂ > 3

This plot depicts the 'merge-split' type of interaction.

KdV (p = 1) colliding wave solution with speed ratio c₁ / c₂ < 3

This plot depicts the 'bouce-exchange' type of interaction.

Remarkably, in all collisions, the only remnant of the nonlinear interaction is a shift in the positions of the two waves.

Colliding wave solutions of the GKdV-p equation for p = 2, 3, 4

A natural question is whether colliding wave solutions of all GKdV-p equations exhibit the same interaction properties as KdV travelling waves.

Collisions of travelling waves with the next weakest nonlinearities p = 2, 3, 4 are depicted in the following plots. The GKdV-2 case is known as the modified KdV (MKdV) equation.

MKdV (p = 2)


GKdV-3


GKdV-4
The MKdV solution exhibits the same properties as the KdV solution, while the GKdV-3 and GKdV-4 solutions develop left-moving tails. The nonlinear interaction for the GKdV-4 solution is so strong that the shorter wave appears to completely stop while the taller wave accelerates.