Propagating stationary profile excitations (solitons) are ubiquitous in Nature. Arising in areas as diverse as telecommunications, hydrodynamics and Space physics, pulse-shaped soliton forms are modeled via generic nonlinear differential equations. In the physics of charged matter (plasma), electrostatic solitons are modelled by means of two theoretical approaches, namely the Sagdeev pseudopotential (SPP) and the reductive perturbation (RPM) method. Both methods come with serious restrictions in terms of the parameter values, which makes their use qualitatively attractive, but rather dubious in terms of quantitative predictions. This project focuses on a novel approach in modeling nonlinear plasma waves, based on a novel hybrid simulation method combining the SPP method with an original kinetic-Vlasov type simulation technique.