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

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

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