Soliton solutions and their dynamics of local and nonlocal (2+1)-dimensional Fokas–Lenells equations

In this work, we obtain one-fold and N-fold Darboux transformation for the integrable (2+1)-dimensional Fokas–Lenells equations by determinant representations. The local and Ablowitz–Musslimani type nonlocal reductions are presented to deduce new integrable systems. A key point for reduced systems is that the special eigenfunctions of spectral problem are used to guarantee the validity of the reduction conditions. Different from the nonlocal (2+1)-dimensional Fokas–Lenells equation, the relation between spectral parameters λ_2j and λ_2j−1 is required in the study of Darboux transformation for local (2+1)-dimensional Fokas–Lenells equation. In view of reduction formulas and different zeed solutions, multi-soliton solutions are derived. We also illustrate one-soliton and two-soliton solutions by plotting their graphs for particular values of the parameters, some of which include bright solitons, periodic waves, shock waves, breathers, dark solitons, antidark solitons, interactions and parallel propagations of mentioned type of waves. Consequently, it is clearly shown that the solutions of nonlocal (2+1)-dimensional Fokas–Lenells equation have new characteristics which differ from the ones of local case.