Darboux transformation and exact soliton solutions of the two-component (2+1)-dimensional Fokas–Lenells equation

Nonlinear wave phenomena have attracted considerable attention since the discovery of solitons by Zabusky and Kruskal. These phenomena play an important role in various areas of science and are described by nonlinear partial differential equations. Among them, integrable models hold a special place due to their ability to admit exact analytical solutions. The Fokas–Lenells (FL) equation is one such model, widely used to describe the propagation of ultrashort optical pulses in nonlinear media. In this work, a two-component (2+1)-dimensional generalization of the FL equation is derived, which takes into account both multimode interactions and multidimensional effects. The corresponding Lax pair is constructed, and the Darboux transformation method is employed to obtain exact one- and two-soliton solutions of the proposed system. The physical parameters of the model — dispersion, nonlinearity, and dimensionality — are shown to significantly affect the structure of the soliton solutions and the modulational instability spectrum. In particular, dispersive terms control the width of the instability band, nonlinear interactions determine the amplitude and growth rate of perturbations, and the additional spatial dimension broadens the instability region through transverse modes. The results emphasize the significance of the proposed model, demonstrate the efficiency of the Darboux method for multicomponent multidimensional systems, and suggest the potential for further exploration of more complex structures such as breathers, rogue waves, and higher-order transformations.