Non-degenerate and degenerate soliton solutions to the semi-discrete vector nonlinear Schrödinger system

In this paper, we derive the non-degenerate and degenerate one- and two-soliton solutions to the semi-discrete vector nonlinear Schrödinger system, which can describe the mean-field waves within the Bose–Einstein condensate system, via Hirota’s bilinear method. The plots with different hump structures for two components are shown under the appropriate restrictions of parameters. Non-degenerate one-soliton solutions that have the double-hump structure and degenerate one-soliton solutions that have the single-hump structure are presented together. Non-degenerate two-soliton solutions can be classified as completely and partially non-degenerate solitons, corresponding to a variety of hump structures for two components. We also show some snapshots of these solitons at different moments. Moreover, a bound state for two-soliton solutions is depicted.