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

The derivation of Boussinesq's type of equations is reexamined for the shallow fluid 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 different signs of the fourth-order dispersion. The head-on collisions among different solitary waves are investigated by means of a conservative difference scheme and their solitonic properties are established, although the inelasticity of collisions is always present.