Compaction - A Soliton with Compact Support

Abstract

Compacton is a soliton with a compact support. Nonlinear dispersion plays a crucial role in its formation The simplest model to see nonlinear dispersion, in action is given by a KdV-like equation, the K(m, n); u _t + (u ^m) _x + (u ⁿ) _xxx = 0, _{m, n > 1}. The compactons are solitary wave solutions of these equations. Their robustnes appears to be very similar to the one observed in completely integrable systems. After “compactons” collision, the interaction site is marked by a small zero-mass ripple which in turn very slowly decomposes into compacton-anticompacton pairs of ever decreasing amplitudes. We have found other equations in one and higher dimensional equations that seem also to have only a finite number of local conservation laws but support compact and semi-compact structures. Notably a fully nonlinear KP equation [u _t + (u ²) _x + (u ²) _xxx ] _x + u _yy + u _zz = 0 may be nontrivially reduced into a K(2, 2) and has a semi-compact decaying solutions