  A novel class of high-order linearly implicit energy-preserving exponential
integrators are proposed for the nonlinear Schr\"odinger equation. We firstly
done that the original equation is reformulated into a new form with a modified
quadratic energy by the scalar auxiliary variable approach. The spatial
derivatives of the system are then approximated with the standard Fourier
pseudo-spectral method. Subsequently, we apply the extrapolation technique to
the nonlinear term of the semi-discretized system and a linearized system is
obtained. Based on the Lawson transformation, the linearized system is
rewritten as an equivalent one and we further apply the symplectic Runge-Kutta
method to the resulting system to gain a fully discrete scheme. We show that
the proposed scheme can produce numerical solutions along which the modified
energy is precisely conserved, as is the case with the analytical solution and
is extremely efficient in the sense that only linear equations with constant
coefficients need to be solved at every time step. Numerical results are
addressed to demonstrate the remarkable superiority of the proposed schemes in
comparison with other high-order structure-preserving method.
