Soliton extraction and geometric analysis for the Hamiltonian amplitude equation

This study presents an integrated analytical and geometric investigation of the Hamiltonian amplitude equation through Lie-point symmetry analysis, conservation-law construction, Hamiltonian–Poisson formulation, and soliton-based numerical verification, together with an exact-solution analysis of its M-truncated form. The mass, momentum, and energy conservation laws are derived using the multiplier method and are independently recovered through an extended Lagrangian formulation and Noether’s theorem, demonstrating the consistency of these approaches. The Lie-point symmetry generators associated with time and space translations, phase rotation, and Galilean invariance are explicitly constructed. The equation is subsequently formulated within a Hamiltonian–Poisson framework, showing that the conserved functionals generate the corresponding continuous symmetry flows through the Poisson bracket. In addition, the rational extended sinh-Gordon equation expansion method is applied to the M-truncated form of the equation to derive several families of exact hyperbolic-function solutions, including dark, singular, mixed bright–dark, and mixed singular soliton structures. The analysis establishes the Lie symmetry, conservation-law, Noether, and Hamiltonian–Poisson structures of the Hamiltonian amplitude equation, while extending its exact-solution treatment to the M-truncated form through distinct soliton families. The derived conservation laws are subsequently verified numerically using bright and dark soliton profiles of the Hamiltonian amplitude equation. The conserved quantities remain invariant to high numerical accuracy over long propagation intervals, with relative errors remaining below 10⁻⁶ . Two-dimensional, three-dimensional, and contour plots further illustrate the localized and shape-preserving profiles of the obtained solutions. These findings confirm the consistency between the analytical and numerical results and clarify the relationships among symmetry transformations, conserved quantities, Hamiltonian structures, and soliton propagation.