Dispersive partial differential equations (PDEs) have important applications in different research areas such as hydrodynamics, optics, plasma physics and medical imaging. In this project these PDEs, including in higher dimensions and in periodic settings, will be studied with a unique innovative combination of analytical and numerical approaches and techniques from the theory of integrable systems, also applied to non-integrable PDEs. The goal is to use the predictive power of numerical techniques for breakthroughs on the analytical side, and analytical insight into the equations to generate innovative numerical schemes able to address challenges in applications numerically. An important question is to what extent the global behavior of solutions to integrable equations or systems holds true for the solutions of non-integrable versions of those equations, for instance Boussinesq systems appearing in the context of water waves, and this also for non-local equations. For integrable equations with global solutions, the long-time behavior is to be determined. Of particular interest are higher-dimensional situations, also in a periodic setting. The to be gained analytical insight and the to be developed numerical algorithms will be directly applicable to Electrical Impedance Tomography, a form of medical imagery also applicable in geology.