Optical solutions and bifurcations of Kudryashov’s arbitrary refractive index along with generalized nonlocal nonlinearities

In this paper, a generalized nonlinear Schrödinger-type equation with higher-order nonlocal nonlinearities and Kudryashov’s arbitrary refractive index is studied. A wide range of physical phenomena, such as nonlinear optics, plasma dynamics, and wave propagation in dispersive media, are modeled by this equation. By applying a traveling wave transformation, the system is reduced to a singular planar dynamical system, which is subsequently regularized to facilitate a comprehensive bifurcation analysis. The equilibrium points are classified based on system parameters, and corresponding phase portraits are constructed to illustrate the qualitative dynamics across various bifurcation scenarios. The unified Riccati equation expansion method and the sine-Gordon expansion method are used to obtain explicit analytical soliton solutions, which are expressed in trigonometric and hyperbolic forms and capture a range of wave structures with different physical properties. Graphic representations in 2D and 3D are presented to illustrate the propagation dynamics. The results contribute to the current solution landscape of nonlocal nonlinear systems and provide new information on the Hamiltonian structure and bifurcation behavior of such singular wave models. To the best of our knowledge, the dynamical classification and exact solutions presented here are new and have not been published before.