<p>The present study conducts a numerical investigation of the classical coupled Jaulent–Miodek (JM) system. The coupled JM system incorporates an energy dependent Schrödinger potential and has applications in fluid dynamics, condensed matter physics, optics, and various engineering systems. We present a geometric meshless approach for the nonlinear coupled JM system in this paper. Pseudo-spectral method is recognized for their high accuracy; however, they face constraints regarding geometric adaptability. The conjunction of radial basis function (RBF) with the pseudo-spectral method effectively addresses this limitation. The Radial basis function Pseudo-spectral method (RBF-PSM) was employed to approximate the coupled JM system by converting it into a set of ordinary differential equations (ODEs). The solution to this system of ODEs has been obtained using the (7,9) Runge–Kutta (RK) method, which is characterized by its seventh order and nine stages. The physical characteristics of these solutions are illustrated through three-dimensional and two-dimensional figures, which aid in understanding the physical phenomena associated with the dynamic models that emerge in mathematical physics. A comparative analysis has also been performed between the solutions derived from the proposed method, the exact solution, and several recently published methods in the literature. Numerical experiments demonstrate that the suggested method is both effective and precise for the JM coupled system.</p>