A focusing and defocusing semi-discrete complex short-pulse equation and its various soliton solutions

摘要

In this paper, we are concerned with a semi-discrete complex short-pulse (sdCSP) equation of both focusing and defocusing types, which can be viewed as an analogue to the Ablowitz-Ladik lattice in the ultra-short-pulse regime. By using a generalized Darboux transformation method, various soliton solutions to this newly integrable semi-discrete equation are studied with both zero and non-zero boundary conditions. To be specific, for the focusing sdCSP equation, the multi-bright solution (zero boundary conditions), multi-breather and high-order rogue wave solutions (non-zero boundary conditions) are derived, while for the defocusing sdCSP equation with non-zero boundary conditions, the multi-dark soliton solution is constructed. We further show that, in the continuous limit, all the solutions obtained converge to the ones for its original CSP equation (Ling et al. 2016 Physica D327, 13-29 (doi:10.1016/j.physd.2016.03.012); Feng et al. 2016 Phys. Rev. E93, 052227 (doi:10.1103/PhysRevE.93.052227)).

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