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

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

• when a < 0

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

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