# 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*^{p}u_{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
3

*c* : 2 / √

*c*.

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

This plot depicts the overlay of two travelling waves with speeds

*c*_{1} >

*c*_{2}.

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

*c*_{1} and

*c*_{2}
are the respective speeds of the faster and slower waves.
When the speed ratio is

*c*_{1} /

*c*_{2} > 3,
the waves first merge and then split apart into their previous shape and speed.
In contrast, when the speed ratio is

*c*_{1} /

*c*_{2} < 3,
the waves bounce and exchange both their shape and speed.

### KdV (*p* = 1) colliding wave solution with speed ratio *c*_{1} / *c*_{2} > 3

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

### KdV (*p* = 1) colliding wave solution with speed ratio *c*_{1} / *c*_{2} < 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.