Bifurcation analysis and exploration of new optical soliton solutions in parabolic law medium with weak non-local nonlinearity

Abstract In the present study, we examine how optical solitons behave in a synthetic nonlocal nonlinear medium. An analytical investigation is conducted into the nonlinear dynamical model that represents the propagation of optical solitons in weakly nonlocal nonlinear media with parabolic law nonlinearity. Due to its capacity to explain intricate physical phenomena and display dynamic structures of localized wave solutions, this equation has attracted a lot of attention. The Khater method and the $$\left( {\frac{1}{{G'}}} \right)$$ -expansion approach are the two analytical techniques used to obtain a wide range of soliton solutions, including mixed, kink and anti-kink profiles. The obtained soliton solutions indicate that the model can describe stable pulse transmission, optical switching fronts, and nonlinear wave modulation. These results are potentially useful for applications in optical communication systems, pulse shaping, and nonlinear signal processing. We also offer a thorough bifurcation analysis that goes beyond solution formation to identify distinct dynamical regimes that mark the boundaries of stability, instability, and periodicity transitions. To investigate the dynamical behavior of the solutions, graphical representations and numerical simulations are offered. This study provides a foundation for further investigation and illustration of other dynamical features present in the physical processes. To the best of our knowledge, the results reported in this work are completely novel and have not been examined in previous literature.