On a model equation of traveling and stationary compactons

We introduce and study a new class of nonlinear dispersive equations: ut+(um)x+[Q(u,ux,uxx)]x=0ut+(um)x+[Q(u,ux,uxx)]x=0, where Q(u,ux,uxx)=q0(u,ux)uxx+q1(u,ux)ux2 is the dispersive flux with typical q′sq′s being monomials in u   and uxux (which amalgamates all KdV type equations with a monomial nonlinear dispersion) and show that it admits either traveling or stationary compactons. In the second case initial datum given on a compact support evolves into a sequence of stationary compactons, with the spatio-temporal evolution being confined to the initial support. We also discuss an N  -dimensional extension ut+(um)x+[ua(∇u)2κ∇2ub]x=0ut+(um)x+[ua(∇u)2κ∇2ub]x=0 which induces N-dimensional compactons convected in x-direction. Two families of explicit solutions of N-dimensional compactons are also presented.