Modified Fokas–Lenells Equation: Self-Consistent Sources and Soliton Solutions of the Spin and (2+1)-Dimensional Models

Nonlinear evolution equations play a key role in modeling various physical processes, such as wave propagation in nonlinear optical and hydrodynamic media, as well as in the dynamics of plasma and quantum systems. In this paper, we study an integrable generalization of the nonlinear Schrödinger equation: the Fokas–Lenells (FL) equation. We derive a new (1+1)-dimensional FL equation with self-consistent sources, which enables modeling the interaction of solitons with external disturbances within the framework of integrable systems. For the frist time, we obtain, two distinct types of solutions for the spin system of the FL equation, namely, a traveling wave and a one-soliton solution, derived using the Darboux transformation (DT). We also construct exact one-soliton and two-soliton solutions for the (2+1)-dimensional FL equation using the DT. These results advance analytical methods in the theory of integrable nonlinear systems, including spin models widely used to describe magnetic, quantum, and soliton phenomena. We illustrate the dynamics of the solutions graphically.