  All complex fluid motions, such as transition and turbulence, obeying the
Navier-Stokes equations are non-linear phenomena. Some aspects of the
non-linear terms of these equations are not well understood and are, in fact,
misunderstood. The one-dimensional Kuramoto-Sivashinsky (KS) equation is used
as a simple model non-linear partial differential equation to show some
essential functions of its non-linear term and its consequences, which, we
believe, are shared with other non-linear partial differential equations. We
show that solutions of nonlinear partial differential equations above their
critical parameters may be linearly stable, but are nonlinearly unstable. No
stable solution exists above the critical parameter, contrary to the prediction
of the linear-stability analysis. This is because a linearly stable disturbance
can transfer energy simultaneously, not necessarily in cascade from small wave
numbers to large wave numbers. An initial disturbance can breed its entire
harmonics simultaneously. Second, I show that a long-time numerical chaotic
solution cannot be achieved by a discrete numerical method.
