Dark and antidark solitons on continuous and doubly periodic backgrounds in the space-shifted nonlocal nonlinear Schrödinger equation

In the framework of the space-shifted nonlocal nonlinear Schrödinger equation, the dark and antidark soliton solutions on both continuous and doubly periodic backgrounds are studied in detail. By performing the long-time asymptotic analysis of the analytic solutions, three key properties of these solitons on continuous background are thoroughly investigated: (i) The explicit classifications of solitons in accordance with their states are provided; (ii) The essential distinctions between these nonlocal solitons and the associated local solitons are identified by examining the correlations of their amplitudes and velocities; (iii) The role of the shifting factor x0 in these solitons is elucidated. A particular solution family showcasing multiple periodic waves with periodicity along both the x and t dimensions is investigated. Specifically, these doubly periodic waves are transformed into the algebraic solitons as their periods tend to infinity. The asymptotic analysis concerning t→±∞ is also conducted for the dark and antidark solitons on doubly periodic background, and reveal two unique soliton properties: (i) The specific parameter governing the amplitudes of periodic waves, unrelated to soliton solutions on continuous background, can convert the solitons from dark states to antidark states or antidark states to dark states; (ii) The amplitudes of the solitons exhibit irregular oscillatory periodicity along t, rather than regular periodicity.