  Linearisation is often used as a first step in the analysis of nonlinear
initial boundary value problems. The linearisation procedure frequently results
in a confusing contradiction where the nonlinear problem conserves energy and
has an energy bound but the linearised version does not (or vice versa). In
this paper we attempt to resolve that contradiction and relate nonlinear energy
conserving and bounded initial boundary value problems to their linearised
versions and the related dual problems. We start by showing that a specific
skew-symmetric form of the primal nonlinear problem leads to energy
conservation and a bound. Next, we show that this specific form together with a
non-standard linearisation procedure preserves these properties for the new
slightly modified linearised problem. We proceed to show that the corresponding
linear and nonlinear dual (or self-adjoint) problems also have bounds and
conserve energy due to this specific formulation. Next, the implication of the
new formulation on the choice of boundary conditions is discussed. A
straightforward nonlinear and linear analysis may lead to a different number
and type of boundary conditions required for an energy bound. We show that the
new formulation shed some light on this contradiction. We conclude by
illustrating that the new continuous formulation automatically lead to energy
stable and energy conserving numerical approximations for both linear and
nonlinear primal and dual problems if the approximations are formulated on
summation-by-parts form.
