Exact Wave Structures and Modulation Instability in the Fokas System

The analytically integrable Fokas system, arising under the slowly varying envelope approximation for weakly nonlinear and weakly dispersive quasi-monochromatic waves, is used to describe pulse propagation in single-mode optical fibers and is investigated here through symbolic computational techniques. This paper establishes multiple families of exact wave solutions through the combined use of the modified simple equation strategy and the generalized exponential rational function technique. These analytical approaches enable the derivation of diverse solitary and periodic wave structures characterized by adjustable parameters that control the amplitude, shape, and propagation dynamics of the waveform. To demonstrate the physical significance of the derived solutions, comprehensive graphical visualizations are provided, highlighting symmetric propagation features and diverse parameter-dependent behaviors of the wave structures. The flexibility of the obtained solution structures allows for a detailed examination of parameter-dependent wave dynamics and waveform evolution within the considered model. Moreover, a detailed modulation instability analysis is carried out to investigate the stability characteristics of continuous-wave solutions in the context of the Fokas system. The results identify parameter regimes associated with stable and unstable wave propagation, thereby enhancing the understanding of nonlinear instability phenomena in integrable optical models. In general, the study contributes new analytical wave structures, stability interpretations, and parametric insights that extend the applicability of the Fokas system in nonlinear wave theory and optical physics.