Higher-order nonlinear Schrödinger equation: Conservation laws, soliton dynamics, and bifurcation analysis

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The higher-order nonlinear Schrödinger equation (h-oNLSE) with cubic–quintic nonlinearity governs the propagation of ultrashort optical pulses in nonlinear fiber systems and plasma environments, where higher-order dispersive and nonlinear perturbations crucially affect pulse stability and shape. Despite extensive studies, the interplay of cubic–quintic nonlinearities with higher-order effects remains insufficiently characterized. In this work, we develop a generalized analytical framework based on a Modified Sardar Sub-Equation Method (mSSEM) to construct new classes of exact solutions to the higher-order cubic–quintic NLSE. This approach systematically uncovers diverse nonlinear waveforms, including previously inaccessible bright and dark solitons, periodic states, and singular structures. Importantly, our results reveal how higher-order dispersion and nonlinear contributions reshape amplitude-phase coupling and stability regimes, offering predictive insights into ultrafast pulse dynamics. By bridging analytical theory with experimentally relevant scenarios in optics and plasma physics, these findings extend the fundamental solution landscape of the cubic–quintic NLSE and establish a versatile methodology applicable across nonlinear evolution equations in applied mathematics and wave science. For the (h-oNLSE) with cubic–quintic nonlinearity, the associated conservation laws have been identified. The analysis confirms the presence of three fundamental conserved quantities corresponding to the bright soliton solutions of the equation. The essential features and physical significance of the conservation laws are discussed, highlighting their role in ensuring the stability and persistence of soliton structures in nonlinear optical media.

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