Geometric admissibility conditions for travelling-wave solitons in the Kuralay-IIA equation

Travelling-wave solutions of nonlinear wave equations describe localized coherent structures in dispersive media, yet the conditions under which such waves are physically admissible are often implicitly assumed. In this work, we investigate this issue for the Kuralay-IIA equation from a geometric dynamical-systems perspective. A travelling-wave reduction converts the original coupled system into a planar autonomous Hamiltonian system governed by an effective potential, where consistency between phase modulation and amplitude dynamics imposes an intrinsic admissibility constraint linking the wave number, frequency, and propagation velocity. Under this constraint, physically admissible travelling waves correspond to phase-space trajectories lying on fixed energy levels. We further introduce an orbit-background consistency criterion, requiring that the asymptotic wave amplitude coincide with an equilibrium of the effective potential. Within this framework, bright solitons arise as homoclinic separatrices attached to saddle equilibria, dark solitons correspond to heteroclinic connections on nonzero backgrounds, and periodic travelling waves are associated with closed orbits surrounding center-type equilibria. The admissible wave behaviour is therefore determined by the geometry of the effective potential rather than by a priori waveform assumptions. These results provide a unified and physically transparent interpretation of travelling-wave admissibility in the Kuralay-IIA equation and offer a general geometric framework applicable to other nonlinear wave equations admitting Hamiltonian travelling-wave reductions.