Discrete conservation laws and the convergence of long time simulations of the mkdv equation

Pseudospectral collocation methods and finite difference methods have been used for approximating an important family of soliton like solutions of the mKdV equation. These solutions present a structural instability which make difficult to approximate their evolution in long time intervals with enough accuracy. In this scenario the numerical schemes which preserve the discrete invariants related to some conservation laws of this equation guarantee better results than the methods which only take care of a high consistency order. Pseudospectral spatial discretization appear as the most robust schemes but finite difference schemes admit an easy formulation for the conservation of the invariants.