Summary: Two local conservation laws of the $K(m,n)$ equation ${u}_{t}\pm {\left({u}^{m}\right)}_{x}+{\left({u}^{n}\right)}_{xxx}=0$ are used to define two Lagrange-type transformations into mass and momentum space. These mappings help to identify new integrable cases $\left(K\right(-1,-2)$, $K(-2,-2)$, $K(\frac{3}{2},-\frac{1}{2}))$, transform conventional solitary waves into compactons – solitary waves on a compactum – and relate certain soliton-carrying systems with compacton-carrying systems. Integrable equations are transformed into new integrable equations, and interaction of $N$-solitons of the $m$-KdV $(m=3,n=1)$ is thus projected into an interaction in a compact domain from which $N$ ordered stationary compactons emerge. The interaction of traveling compactons is the image of super-imposed equilibria of the corresponding soliton equation. For $m=n+2$, the potential form of the $K(m,n)$ equation may also be cast into a conserved form and thus transformed, yielding generalized Dym and Wadati equations and two new integrable cases. It is shown that ${r}_{t}+{(1-{r}^{2})}^{3/2}{({r}_{xx}+r)}_{x}=0$ is integrable and supports compact kinks.

##### MSC:

35Q53 | KdV-like (Korteweg-de Vries) equations |

35Q51 | Soliton-like equations |