Dynamics and stability of solitons in the higher-order dispersive cubic-quintic nonlinear schrödinger equation: conservation laws and applications

This study investigates the construction of multi-peak solitons and various other wave solutions of a higher-order nonlinear Schrödinger equation incorporating cubic-quintic dispersion, using the modified exponential rational function method. The considered model plays a crucial role across multiple scientific and engineering disciplines, including quantum mechanics, transmission lines, oceanography, and optical fiber communications. By accounting for higher-order dispersion and nonlinear effects, particularly the cubic-quintic terms, the model provides a more accurate description of the propagation dynamics of ultra-short pulses in optical fibers and oceanic environments. A diverse class of soliton solutions is derived, encompassing bright, dark, trigonometric, hyperbolic, rational, exponential, and singular forms. These results contribute to a deeper understanding of nonlinear wave interactions and complex oceanic phenomena in both shallow and deep-water contexts. Conservation laws for energy, momentum, and mass are established to ensure the physical validity of the obtained solutions, while stability analysis confirms their robustness and reliability. Graphical illustrations further reveal the influence of key parameters on soliton dynamics, offering valuable physical insights. This study addresses existing gaps in the analytical treatment of higher-order dispersive cubic-quintic NLSE by deriving new classes of exact soliton solutions and examining their stability and conservation properties, thereby contributing to a deeper understanding of nonlinear wave dynamics in optical and physical systems. Overall, the findings highlight the efficiency of the mERFM and its potential applicability in solving a broad range of nonlinear mathematical and physical models.