Analysis of Optical Bi-wave Solutions in a Two-mode Model Arising from the Unstable Schr?dinger Equation

In this study, we explore the essential conditions for the existence of optical solitary solutions within a novel two-mode extension derived from the unstable nonlinear Schr?dinger equation. This extendedmodel captures the dynamics of symmetric bi-waves in diverse complex media, ranging from optics and plasma to fluid dynamics and electrical engineering. The propagated bi-wave experience influences from three key factors: nonlinearity, linearity-dispersion, and phase velocity. Specifically, the wave amplitudes (modes) are linked to the interplay between nonlinearity and dispersion parameters, while the wave widths are governed by the phase velocity parameter. As the presented model is introduced here for the first time, we investigate some of its solutions using effective approaches, including the Kudryashov-expansion and modified rational sine-cosine methods. Additionally, we complement our analysis with graphical representations to gain fresh insights into the behavior of the new two-mode model. We anticipate that the findings presented in thiswork will contribute to a deeper understanding of bi-wave propagation phenomena across various research domains.