A complete list of conservation laws for non-integrable compacton equations of $K(m,m)$ type

TL;DR

This paper classifies all conservation laws and symmetries for a class of generalized K(m,m) equations, revealing only a few symmetries and conservation laws for non-integrable cases, and confirming known results for specific equations.

Contribution

It provides a complete description of conservation laws and symmetries for generalized K(m,m) equations, extending previous results and identifying exceptional cases.

Findings

Only three symmetries for most cases, including translations and scaling.

Four nontrivial conservation laws identified, including energy conservation.

Confirmed that K(2,2) has exactly four conservation laws as previously found.

Abstract

In 1993, P. Rosenau and J. M. Hyman introduced and studied Korteweg-de-Vries-like equations with nonlinear dispersion admitting compacton solutions, $u_t+D_x^3(u^n)+D_x(u^m)=0$, $m,n>1$, which are known as the $K(m,n)$ equations. In the present paper we consider a slightly generalized version of the $K(m,n)$ equations for $m=n$, namely, $u_t=aD_x^3(u^m)+bD_x(u^m)$, where $m,a,b$ are arbitrary real numbers. We describe all generalized symmetries and conservation laws thereof for $m\neq -2,-1/2,0,1$; for these four exceptional values of $m$ the equation in question is either completely integrable ($m=-2,-1/2$) or linear ($m=1$) or trivial ($m=0$). It turns out that for $m\neq -2,-1/2,0,1$ there are only three symmetries corresponding to $x$- and $t$-translations and scaling of $t$ and $u$, and four nontrivial conservation laws, one of which expresses the conservation of energy, and the other three are associated with the Casimir functionals of the Hamiltonian operator $\mathfrak{D}=aD_x^3+bD_x$ admitted by our equation. Our result, \textit{inter alia}, provides a rigorous proof of the fact that the K(2,2) equation has just four conservation laws found by P. Rosenau and J. M. Hyman.