  In this paper, we present a novel strategy to systematically construct
linearly implicit energy-preserving schemes with arbitrary order of accuracy
for Hamiltonian PDEs. Such novel strategy is based on the newly developed
exponential scalar variable (ESAV) approach that can remove the
bounded-from-blew restriction of nonlinear terms in the Hamiltonian functional
and provides a totally explicit discretization of the auxiliary variable
without computing extra inner products, which make it more effective and
applicable than the traditional scalar auxiliary variable (SAV) approach. To
achieve arbitrary high-order accuracy and energy preservation, we utilize the
symplectic Runge-Kutta method for both solution variables and the auxiliary
variable, where the values of internal stages in nonlinear terms are explicitly
derived via an extrapolation from numerical solutions already obtained in the
preceding calculation. A prediction-correction strategy is proposed to further
improve the accuracy. Fourier pseudo-spectral method is then employed to obtain
fully discrete schemes. Compared with the SAV schemes, the solution variables
and the auxiliary variable in these ESAV schemes are now decoupled. Moreover,
when the linear terms are of constant coefficients, the solution variables can
be explicitly solved by using the fast Fourier transform. Numerical experiments
are carried out for three Hamiltonian PDEs to demonstrate the efficiency and
conservation of the ESAV schemes.
