Unveiling new quiescent dark and singular solitary solutions of the Fokas–Lenells equation

Abstract This paper investigates novel quiescent solitary waves in the Fokas–Lenells equation, a vital and versatile model in nonlinear dynamics used to describe wave propagation phenomena in optical fibers, plasma physics, and fluid systems. The Fokas–Lenells equation is notable for its ability to capture nonlinear and dispersive effects, making it a cornerstone for understanding complex wave behaviors in various physical contexts. By employing advanced analytical techniques, we derive new classes of dark and singular solitons. These solutions, previously unexplored in the context of the Fokas–Lenells equation, significantly enhance the repertoire of soliton solutions available for this nonlinear system. The novelty of this study lies in the systematic application of these diverse methods, providing a comparative analysis that underscores their respective strengths and practical advantages in addressing nonlinear evolution equations. The derived solutions not only demonstrate the efficacy and versatility of these techniques, but also reveal intricate dynamics inherent to the Fokas–Lenells equation. The physical significance of the newly obtained solutions is thoroughly examined, highlighting their relevance in describing interactions of waves in nonlinear optical and fluid systems. Such insights are crucial for advancing theoretical models and practical applications in fields where nonlinear wave phenomena are prominent. This study contributes to the ongoing development of soliton theory by extending the analytical framework used for solving the Fokas–Lenells equation, deepening the theoretical understanding of this model, and laying the groundwork for future explorations into soliton behaviors and nonlinear science.