  A novel class of high-order linearly implicit energy-preserving integrating
factor Runge-Kutta methods are proposed for the nonlinear Schr\"odinger
equation. Based on the idea of the scalar auxiliary variable approach, the
original equation is first reformulated into an equivalent form which satisfies
a quadratic energy. The spatial derivatives of the system are then approximated
with the standard Fourier pseudo-spectral method. Subsequently, we apply the
extrapolation technique/prediction-correction strategy to the nonlinear terms
of the semi-discretized system and a linearized energy-conserving system is
obtained. A fully discrete scheme is gained by further using the integrating
factor Runge-Kutta method to the resulting system. We show that, under certain
circumstances for the coefficients of a Runge-Kutta method, 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 existing structure-preserving schemes.
