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In the paper, the conservative Fourier spectral scheme is presented for the coupled Schrödinger–Boussinesq equations. We apply the Fourier collocation scheme to spatial derivatives and the Crank–Nicolson scheme to the system in time direction, respectively. We find that the scheme can preserve mass and energy conservation laws. Moreover, the existence, uniqueness, stability and convergence of the scheme are discussed, and it is shown that the scheme is of the accuracy \(O(\tau^{2}+J^{-r})\). The numerical experiments are given to show that verify the correctness of theoretical results and the efficiency of the scheme.

Conservative Fourier spectral scheme for the coupled Schrödinger–Boussinesq equations