Well-posed Boussinesq paradigm with purely spatial higher-order derivatives.

The derivation of Boussinesq's type of equations is revisited for the shallow uid layers and nonlinear atomic chains. It is shown that the linearly stable equation with purely spatial derivatives representing dispersion must be of sixth order. The corresponding conservation and balance laws are derived. The shapes of solitary stationary waves are calculated numerically for diierent signs of the fourth{order dispersion. The head{on collisions among diierent solitary waves are investigated by means of a conservative diierence scheme and their solitonic properties are established, although the inelasticity of collisions is always present.