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Soliton Geometry, Center-Manifold Dynamics, and Nonlinear Phase-Space Structure of the Kuralay Equation
Subodh Barik, Akash Dehury and Sidheswar Behera

In this work, the nonlinear wave dynamics of the Kuralay equation are investigated through a combination of traveling-wave reduction, phase-space analysis, and Hirotas bilinear formalism. The governing system is reduced to a four-dimensional autonomous dynamical system whose equilibrium structure and center-manifold geometry reveal bounded invariant trajectories supporting coherent wave motion. Exact analytical solutions, including one-soliton, two-soliton, and multi-soliton structures, are systematically constructed using Hirota’s method. The interaction of multi-soliton solutions is shown to be elastic, with amplitudes preserved and finite phase shifts arising from nonlinear interaction. The phase portraits provide a geometric interpretation of the balance between dispersion and nonlinear coupling that sustains stable localized waveforms. These results highlight the rich soliton-supporting structure of the Kuralay model and demonstrate its relevance to nonlinear wave propagation in coupled dispersive media such as nonlinear optics and plasma systems.

Keywords: The Hirota Bilinear method, phase portrait analysis, stability analysis, Kuralay equation