On a model equation of traveling and stationary compactons

We introduce and study a new class of nonlinear dispersive equations: u(t) + (u(m))(x) + [Q(u, u(x), u(xx))](x) = 0, where Q(u, u(x), u(xx)) q(0)(u, u(x))u(xx) + q(1)(u, u(x))u(x)(2) is the dispersive flux with typical q's being monomials in u and u(x) (which arnalgamates 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 u(1) + (u(m))(x) + [u(a)(del u)(2 kappa del 2) u(b)](x) = 0 which induces N-dimensional compactons convected in x-direction. Two families of explicit solutions of N-dimensional compactons are also presented. (c) 2006 Elsevier B.V. All rights reserved.