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A novel class of high-order linearly implicit energy-preserving integratingfactor Runge-Kutta methods are proposed for the nonlinear Schr\"odingerequation. Based on the idea of the scalar auxiliary variable approach, theoriginal equation is first reformulated into an equivalent form which satisfiesa quadratic energy. The spatial derivatives of the system are then approximatedwith the standard Fourier pseudo-spectral method. Subsequently, we apply theextrapolation technique/prediction-correction strategy to the nonlinear termsof the semi-discretized system and a linearized energy-conserving system isobtained. A fully discrete scheme is gained by further using the integratingfactor Runge-Kutta method to the resulting system. We show that, under certaincircumstances for the coefficients of a Runge-Kutta method, the proposed schemecan produce numerical solutions along which the modified energy is preciselyconserved, as is the case with the analytical solution and is extremelyefficient in the sense that only linear equations with constant coefficientsneed to be solved at every time step. Numerical results are addressed todemonstrate the remarkable superiority of the proposed schemes in comparisonwith other existing structure-preserving schemes.

High-order linearly implicit structure-preserving exponential integrators for the nonlinear Schr\"odinger equation