Nonlinear dispersion and compact structures.

Abstract

Relaxing the distinguished ordering underlying the derivation of soliton supporting equations leads to new equations endowed with nonlinear dispersion crucial for the formation and coexistence of compactons, solitons with a compact support, and conventional solitons. Vibrations of the anharmonic mass-spring chain lead to a new Boussinesq equation admitting compactons and compact breathers. The model equation ${u}_{t}+{[\frac{\ensuremath{\delta}u+3\ensuremath{\gamma}{u}^{2}}{2+{u}^{1\ensuremath{-}\ensuremath{\omega}}{({u}^{\ensuremath{\omega}}{u}_{x})}_{x}}]}_{x}+\ensuremath{\nu}{u}_{\mathrm{txx}}=0(\ensuremath{\omega},\ensuremath{\nu},\ensuremath{\delta},\ensuremath{\gamma} \mathrm{const})$ admits compactons and for $2\ensuremath{\omega}=\ensuremath{\nu}\ensuremath{\gamma}=1$ has a bi-Hamiltonian structure. The infinite sequence of commuting flows generates an integrable, compacton's supporting variant of the Harry Dym equation.