Geometric analysis and nonlinear wave propagation in dispersive and inhomogeneous media with Hamiltonian flows

In this work, we investigate the symmetry structure, conservation laws, and Hamiltonian formulation of the Hirota equation, which represents a higher-order integrable extension of the nonlinear Schrödinger equation. Using the multiplier method, we derive conserved quantities associated with power, momentum, and energy in a systematic and direct manner. The obtained conservation laws are shown to be in complete agreement with those derived from Noether’s theorem through a generalized Lagrangian formulation. The Lie-point symmetry generators of the Hirota equation are constructed explicitly, including time and space translations, phase rotation, Galilean invariance, and scaling symmetry. We establish a rigorous connection between Lie symmetries, Noether symmetries, and Hamiltonian flows by formulating the equation within a Hamiltonian–Poisson framework. The conserved quantities are shown to generate continuous symmetry transformations through the Poisson bracket, revealing the underlying geometric structure of the model. Furthermore, exact one-soliton solutions are employed to evaluate the conserved quantities explicitly, and their invariance is verified numerically over time. On the other hand, using the rational extended sinh–Gordon technique, single rational dark, single rational singular, mixed rational dark–bright, mixed rational singular, mixed rational dark–bright-singular-periodic and mixed singular-periodic-singular solitons are successfully extracted. Graphical illustrations, including three-dimensional, and contour plots, demonstrate the stability and shape-preserving nature of the soliton solutions. The results confirm the complete integrability of the Hirota equation and provide a unified framework linking symmetry analysis, conservation laws, and soliton dynamics.