  We consider a wide class of semi linear Hamiltonian partial differential
equa- tions and their approximation by time splitting methods. We assume that
the nonlinearity is polynomial, and that the numerical tra jectory remains at
least uni- formly integrable with respect to an eigenbasis of the linear
operator (typically the Fourier basis). We show the existence of a modified
interpolated Hamiltonian equation whose exact solution coincides with the
discrete flow at each time step over a long time depending on a non resonance
condition satisfied by the stepsize. We introduce a class of modified splitting
schemes fulfilling this condition at a high order and prove for them that the
numerical flow and the continuous flow remain close over exponentially long
time with respect to the step size. For stan- dard splitting or
implicit-explicit scheme, such a backward error analysis result holds true on a
time depending on a cut-off condition in the high frequencies (CFL condition).
This analysis is valid in the case where the linear operator has a discrete
(bounded domain) or continuous (the whole space) spectrum.
