Rosenau–Hyman equation

The Rosenau–Hyman equation or K(n,n) equation is a KdV-like equation having compacton solutions. This nonlinear partial differential equation is of the form

$u_t+a(u^n)_x+(u^n)_{xxx}=0. \,$

The equation is named after Philip Rosenau and James M. Hyman, who used in their 1993 study of compactons.

The K(n,n) equation has the following traveling wave solutions:

• when a > 0

$u(x,t)= \left( \frac{2cn}{a(n+1)} \sin^2 \left(\frac{n-1}{2n}\sqrt{a}(x-ct+b)\right)\right)^{1/(n-1)},$

• when a < 0

$u(x,t)=\left( \frac{2cn}{a(n+1)}\sinh^2\left(\frac{n-1}{2n}\sqrt{-a}(x-ct+b)\right)\right)^{1/(n-1)},$

$u(x,t)= \left( \frac{2cn}{a(n+1)} \cosh^2 \left(\frac{n-1}{2n}\sqrt{-a}(x-ct+b)\right)\right)^{1/(n-1)}.$