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 and the Degasperis–Procesi 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. [Camassa & Holm 1993]

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. [Degasperis, Holm & Hone 2002] 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 $u(x,t)\; =\; c\; ,\; e^\{-|x-ct$,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, $m\; =\; u\; -\; u\_\{xx\}\; =\; c\; ,\; delta(x-ct)$.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 $G(x)\; =\; exp(-|x|)$, and use this to rewrite the PDE as a (nonlocal) hyperbolic conservation law:

:$partial\_t\; u\; +\; partial\_x\; left\; [frac\{u^2\}\{2\}\; +\; frac\{G\}\{2\}\; *\; left(frac\{b\; u^2\}\{2\}\; +\; frac\{(3-b)\; u\_x^2\}\{2\}\; ight)\; ight]\; =\; 0.$

(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. [Constantin & McKean 1999 (who treat the Camassa–Holm case "b" = 2; the general case is very similar)]

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 KdV 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

:$dot\{x\}\_k\; =\; sum\_\{i=1\}^n\; m\_i\; e^\{-|x\_k-x\_i,qquaddot\{m\}\_k\; =\; (b-1)\; sum\_\{i=1\}^n\; m\_k\; m\_i\; sgn(x\_k-x\_i)\; e^\{-|x\_k-x\_iqquad(k\; =\; 1,dots,n)$

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 $exp(-|x|)$ at "x" = 0 as being zero.This gives the following convenient shorthand notation for the system:

:$dot\{x\}\_k\; =\; u(x\_k),qquaddot\{m\}\_k\; =\; -(b-1)\; m\_k\; u\_x(x\_k)qquad(k\; =\; 1,dots,n).$

The first equation provides some useful intution 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 [Beals, Sattinger & Szmigielski 2000 (where a different normalization and sign convention is used)]

:

where $a\_k(t)\; =\; a\_k(0)\; e^\{t/lambda\_k\}$, and where the 2"n" constants $a\_k(0)$ and $lambda\_k$ are determined from initial conditions. The general solution for arbitrary "n" can be expressed in terms of symmetric functions of $a\_k$ and $lambda\_k$. The general "n"-peakon solution in the Degasperis–Procesi case "b" = 3 is similar in flavour, although the detailed structure is more complicated. [Lundmark & Szmigielski 2005]

Notes

References

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