Soliton, multistability, and chaotic dynamics of the higher-order nonlinear Schrödinger equation

This work finds new exact soliton solutions to the fractional space–time higher-order nonlinear Schrödinger equation, describing how tiny pulses move through a nonlinear system. First, we transform this nonlinear fractional differential equation into an ordinary differential framework using the beta derivative and a traveling wave transformation. Then, we find analytical solutions using the unified solver method. Along with this, a thorough stability analysis is done using the Hamiltonian technique. Afterward, we study the chaotic analysis of the stated model using planner dynamics and show two- and three-dimensional phase illustrations, Lyapunov exponents, Poincaré maps, bifurcation figures, fractal dimensions, strange attractors, recurrence plots, and return maps as graphical representations regarding this chaotic analysis. Finally, we ensure that these precise solitons provide the internal complex image of wave travel.