Analytical wave solutions and bifurcation analysis for optical pulse propagation in resonant nonlinear Schrödinger equation using improved modified extended tanh function method

Abstract Background The study of optical pulse propagation through nonlinear media plays a fundamental role in the development of modern fiber-optic communication systems. Understanding the complex interaction between dispersion, cubic–quintic nonlinearities, resonance, and self-steepening effects is essential for describing the evolution and stability of optical pulses in nonlinear fibers. The resonant nonlinear Schrödinger equation that includes these effects provides a reliable mathematical model to analyze and predict nonlinear wave dynamics in optical and plasma environments. Methods In this research, the improved modified extended tanh function method is employed to obtain exact analytical wave solutions of the resonant cubic–quintic nonlinear Schrödinger equation that contains derivative nonlinearities. The proposed analytical procedure transforms the governing nonlinear partial differential equation into an ordinary differential form by using a traveling-wave transformation. The balance principle between the nonlinear and dispersive terms is then applied to determine the structure of the assumed solution. By substituting this structure into the reduced equation, a system of algebraic equations is generated and solved symbolically to determine the unknown constants. The method enables the construction of various exact analytical forms, including solitary, periodic, and singular wave solutions, through the extended Riccati equation. Results A wide variety of analytical solutions are obtained, such as bright solitons, dark solitons, singular solitons, Weierstrass elliptic wave solutions, rational wave solutions, and exponential wave structures. Graphical illustrations in two and three dimensions clearly demonstrate the distinct dynamical behaviors of these solutions and show how variations in parameters influence their amplitude, width, and stability. Furthermore, a detailed bifurcation analysis is carried out to investigate the qualitative characteristics and stability of the derived solutions. The results reveal the existence of saddle and center equilibrium points, highlighting the dependence of wave stability on nonlinear and dispersive coefficients. Conclusions This study presents, for the first time, the application of the improved modified extended tanh function method to the resonant cubic–quintic nonlinear Schrödinger equation with derivative nonlinearities. The findings confirm that the proposed method provides a systematic and efficient analytical framework for generating exact solutions that accurately capture the complex dynamics of nonlinear wave propagation in optical and plasma media. The newly discovered families of soliton and periodic solutions enhance the theoretical understanding of nonlinear wave behavior and contribute significantly to the field of nonlinear optical communication.