This study investigates soliton dynamics and chaotic analysis of the (1 + 1)-dimensional Hamilton amplitude (HA) model, which arises in mathematical physics. Using two improved ways, namely the improved Kudryashov and the improved \(\left(\frac{{G}{\prime}}{G}\right)\) -expansion approaches, we get periodic waves, periodic breathers, periodic waves with local breathers, double periodic waves, multiple breathers, multiple periodic waves, and single breather waves. To visually confirm these solutions analytically and numerically, we employ 3D, 2D, and overlapping phenomena by scatter plots. We also use various tools to examine chaotic dynamics in the governing model, including power spectra, return maps, bifurcation plots, Lyapunov exponents, fractal dimensions, recurrence plots, basins of attraction, and strange attractor plots. The outcomes show that our employed scheme is more effective, reliable, and simpler than existing schemes.
