### Peakon

In the theory of integrable systems, a **peakon** ("peaked soliton") is a soliton with discontinuous first derivative; the wave profile is shaped like the graph of the function $e^\{-|x|\}$. Some examples of non-linear partial differential equations with (multi-)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.^{[1]}

## Contents

## A family of equations with peakon solutions

The primary example of a PDE which supports peakon solutions is

- $$

u_t - u_{xxt} + (b+1) u u_x = b u_x u_{xx} + u u_{xxx}, \,

where $u(x,t)$ is the unknown function, and *b* is a parameter.^{[2]}
In terms of the auxiliary function $m(x,t)$ defined by the relation $m\; =\; u-u\_\{xx\}$, the equation takes the simpler form

- $$

m_t + m_x u + b m u_x = 0. \,

This equation is integrable for exactly two values of *b*, namely *b* = 2 (the Camassa–Holm equation) and *b* = 3 (the Degasperis–Procesi equation).

## The single peakon solution

The PDE above admits the travelling wave solution $u(x,t)\; =\; c\; \backslash ,\; e^\{-|x-ct|\}$,
which is a peaked solitary wave with amplitude *c* and speed *c*.
This solution is called a (single) 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 *u*_{x} has a jump discontinuity at the peak,
the second derivative *u*_{xx} must be taken in the sense of distributions and will contain a Dirac delta function;
in fact, $m\; =\; u\; -\; u\_\{xx\}\; =\; c\; \backslash ,\; \backslash delta(x-ct)$.
Now the product $m\; u\_x$ occurring in the PDE seems to be undefined, since the distribution *m* is supported at the very point where the derivative *u*_{x} is undefined. An ad hoc interpretation is to take the value of *u*_{x} at that point to equal the average of its left and right limits (zero, in this case). A more satisfactory way to make sense of the solution is to invert the relationship between *u* and *m* by writing $m\; =\; (G/2)\; *\; u$, where $G(x)\; =\; \backslash exp(-|x|)$, and use this to rewrite the PDE as a (nonlocal) hyperbolic conservation law:

- $$

\partial_t u + \partial_x \left[\frac{u^2}{2} + \frac{G}{2} * \left(\frac{b u^2}{2} + \frac{(3-b) u_x^2}{2} \right) \right] = 0.

(The star denotes convolution with respect to *x*.)
In this formulation the function *u* can simply be interpreted as a weak solution in the usual sense.^{[3]}

## Multipeakon solutions

Multipeakon solutions are formed by taking a linear combination of several peakons, each with its own time-dependent amplitude and position. (This is a very simple structure compared to the multisoliton solutions of most other integrable PDEs, like the Korteweg–de Vries equation for instance.)
The *n*-peakon solution thus takes the form

- $$

u(x,t) = \sum_{i=1}^n m_i(t) \, e^{-|x-x_i(t)|},

where the 2*n* functions $x\_i(t)$ and $m\_i(t)$
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

- $$

\dot{x}_k = \sum_{i=1}^n m_i e^{-|x_k-x_i|}, \qquad \dot{m}_k = (b-1) \sum_{i=1}^n m_k m_i \sgn(x_k-x_i) e^{-|x_k-x_i|} \qquad (k = 1,\dots,n)

is satisfied. (Here sgn denotes the sign function.)
Note that the right-hand side of the equation for $x\_k$ is obtained by substituting $x=x\_k$ in the formula for *u*.
Similarly, the equation for $m\_k$ can be expressed in terms of $u\_x$, if one interprets the derivative of $\backslash exp(-|x|)$ at *x* = 0 as being zero.
This gives the following convenient shorthand notation for the system:

- $$

\dot{x}_k = u(x_k), \qquad \dot{m}_k = -(b-1) m_k u_x(x_k) \qquad (k = 1,\dots,n).

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^{[4]}

- $$

\begin{align} x_1(t) &= \log\frac{(\lambda_1-\lambda_2)^2 (\lambda_1-\lambda_3)^2 (\lambda_2-\lambda_3)^2 a_1 a_2 a_3}{\sum_{j

where $a\_k(t)\; =\; a\_k(0)\; e^\{t/\backslash lambda\_k\}$, and where the 2

nconstants $a\_k(0)$ and $\backslash lambda\_k$ are determined from initial conditions. The general solution for arbitraryncan be expressed in terms of symmetric functions of $a\_k$ and $\backslash lambda\_k$. The generaln-peakon solution in the Degasperis–Procesi caseb= 3 is similar in flavour, although the detailed structure is more complicated.^{[5]}## Notes

## References

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