Analysis of soliton wave structure for coupled Higgs equation via Lie symmetry, Paul Painlevé approach and the Unified method

In the present study, we explore novel analytical solutions to the nonlinear coupled Higgs field equation by leveraging symmetry analysis. Specifically through the identification and utilization of optimal subalgebras, we have derived a set of exact solutions, where some of them describe dark-bright, periodic, and breather soliton solutions, that provide significant insights into the behaviour of the equation. Further, we employ several advanced mathematical techniques, such as the Paul Painlevé analysis, the Unified method, and the extended \(\tanh \) $tanh$ method, to investigate the equation further. We apply the Paul Painlevé analysis to identify the conditions under which the equation provides the soliton solution and provides a broad spectrum of single soliton solutions. Moreover, a periodic soliton solution is identified under the Unified method. Lastly, the extended \(\tanh \) $tanh$ method is employed to extract additional soliton solutions and offer a deeper understanding of the dynamics.