Geometric analysis and nonlinear wave propagation in dispersive and inhomogeneous media with Hamiltonian flows - Bulut, Hasan; Goktas, Fahrettin; Sulaiman, Tukur Abdulkadir; Yusuf, Abdullahi (World Scientific Publ Co Pte Ltd, 2026)
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 Schr & ouml;dinger 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.
Keywords: Hirota equation; soliton dynamics; multiplier; Lie point symmetry; Hamiltonian poisson; conserved quantities