Shock Waves of the Gerdjikov–Ivanov Equation Using the Adomian Decomposition Schemes

Analytical solutions for the complex-valued nonlinear Gerdjikov–Ivanov (GI) equation have been studied extensively using integrability-based methods. In contrast, numerical and semi-analytical exploration remains relatively underdeveloped. Thus, the present study deploys both the traditional Adomian decomposition method (ADM) and its improved version (IADM) to explore the computational relevance of the GI equation to shock waves against a benchmark exact soliton solution. The findings indicate that both methods are effective in addressing the GI equation, with the improved method demonstrating an enhancement in the stability of the convergence under specific conditions. This work offers the first systematic semi-analytic and numerical evaluation of the GI equation, introducing practical implementation guidelines.