Almost structure-preserving analysis for weakly linear damping nonlinear Schrödinger equation with periodic perturbation

• The local energy/momentum losses are deduced theoretically in Hamiltonian framework and this result is verified by the structure-preserving approach.
• The breakup process of the multisoliton state is simulated and the bifurcation of the discrete eigenvalues of associated Zakharov–Shabat spectral problem are obtained.

Exploring the dynamic behaviors of the damping nonlinear Schrödinger equation (NLSE) with periodic perturbation is a challenge in the field of nonlinear science, because the numerical approaches available for damping-driven dynamic systems may exhibit the artificial dissipation in different degree. In this paper, based on the generalized multi-symplectic idea, the local energy/momentum loss expressions as well as the approximate symmetric form of the linearly damping NLSE with periodic perturbation are deduced firstly. And then, the local energy/momentum losses are separated from the simulation results of the NLSE with small linear damping rate less than the threshold to insure structure-preserving properties of the scheme. Finally, the breakup process of the multisoliton state is simulated and the bifurcation of the discrete eigenvalues of the associated Zakharov–Shabat spectral problem is obtained to investigate the variation of the velocity as well as the amplitude of the solitons during the splitting process.