Peakon


In the theory of integrable systems, a peakon is a soliton with discontinuous first derivative; the wave profile is shaped like the graph of the function. Some examples of non-linear partial differential equations with peakon solutions are the Camassa–Holm shallow water wave equation, the Degasperis–Procesi equation and the Fornberg–Whitham equation.
Since peakon solutions are only piecewise differentiable, they must be interpreted in a suitable weak sense.
The concept was introduced in 1993 by Camassa and Holm in the short but much cited paper where they derived their shallow water equation.

A family of equations with peakon solutions

The primary example of a PDE which supports peakon solutions is
where is the unknown function, and b is a parameter.
In terms of the auxiliary function defined by the relation, the equation takes the simpler form
This equation is integrable for exactly two values of b, namely b = 2 and b = 3.

The single peakon solution

The PDE above admits the travelling wave solution,
which is a peaked solitary wave with amplitude c and speed c.
This solution is called a peakon solution,
or simply a peakon.
If c is negative, the wave moves to the left with the peak pointing downwards,
and then it is sometimes called an antipeakon.
It is not immediately obvious in what sense the peakon solution satisfies the PDE.
Since the derivative ux has a jump discontinuity at the peak,
the second derivative uxx must be taken in the sense of distributions and will contain a Dirac delta function;
in fact,.
Now the product occurring in the PDE seems to be undefined, since the distribution m is supported at the very point where the derivative ux is undefined. An ad hoc interpretation is to take the value of ux at that point to equal the average of its left and right limits. A more satisfactory way to make sense of the solution is to invert the relationship between u and m by writing, where, and use this to rewrite the PDE as a hyperbolic conservation law:
In this formulation the function u can simply be interpreted as a weak solution in the usual sense.

Multipeakon solutions

Multipeakon solutions are formed by taking a linear combination of several peakons, each with its own time-dependent amplitude and position.
The n-peakon solution thus takes the form
where the 2n functions and
must be chosen suitably in order for u to satisfy the PDE.
For the "b-family" above it turns out that this ansatz indeed gives a solution, provided that the system of ODEs
is satisfied.
Note that the right-hand side of the equation for is obtained by substituting in the formula for u.
Similarly, the equation for can be expressed in terms of, if one interprets the derivative of at x = 0 as being zero.
This gives the following convenient shorthand notation for the system:
The first equation provides some useful intuition about peakon dynamics: the velocity of each peakon equals the elevation of the wave at that point.

Explicit solution formulas

In the integrable cases b = 2 and b = 3, the system of ODEs describing the peakon dynamics can be solved explicitly for arbitrary n in terms of elementary functions, using inverse spectral techniques. For example, the solution for n = 3 in the Camassa–Holm case b = 2 is given by
where, and where the 2n constants and are determined from initial conditions. The general solution for arbitrary n can be expressed in terms of symmetric functions of and. The general n-peakon solution in the Degasperis–Procesi case b = 3 is similar in flavour, although the detailed structure is more complicated.