<p>Based on the Darwin’s Hamiltonian, the many-particle quantum hydrodynamics (QHD) is considered. A thorough investigation is conducted into the force field emerging in the associated Euler equation, meticulously tracing the contributions of various terms within Darwin’s Hamiltonian to the Euler equation. The non-linear reductive perturbation technique is employed to derive Korteweg–De Varies (KdV) equation and soliton solution is obtained. The impact of various parameters, e.g., particle concentration, Darwin interaction, and propagation speed of soliton, is studied. We noted significant impacts on both the width and amplitude of the soliton, providing insights into the intricate interaction between quantum relativistic effects and soliton properties.</p>