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 | {\displaystyle 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.
A family of equations with peakon solutions The primary example of a PDE which supports peakon solutions is
u t − u x x t + ( b + 1 ) u u x = b u x u x x + u u x x x , {\displaystyle u_{t}-u_{xxt}+(b+1)uu_{x}=bu_{x}u_{xx}+uu_{xxx},\,}
where u ( x , t ) {\displaystyle u(x,t)} is the unknown function, and b is a parameter. In terms of the auxiliary function m ( x , t ) {\displaystyle m(x,t)} defined by the relation m = u − u x x {\displaystyle m=u-u_{xx}} , the equation takes the simpler form
m t + m x u + b m u x = 0. {\displaystyle m_{t}+m_{x}u+bmu_{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).
Single peakon solution The PDE above admits the travelling wave solution u ( x , t ) = c e − | x − c t | {\displaystyle 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 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, m = u − u x x = c δ ( x − c t ) {\displaystyle m=u-u_{xx}=c\,\delta (x-ct)} . Now the product m u x {\displaystyle mu_{x}} 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 (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 {\displaystyle m=(G/2)*u} , where G ( x ) = exp ( − | x | ) {\displaystyle G(x)=\exp(-|x|)} , and use this to rewrite the PDE as a (nonlocal) hyperbolic conservation law:
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