More than a century ago, Whittaker developed a theory of the 3D Laplace equation and the (2+1)D wave equation based on a general expression of the local solutions. This theory was eclipsed by more general methods of linear PDEs, but at the time it provided a powerful tool for analysis. For completely integrable nonlinear PDEs, a general expression of the local solutions is still a work in progress. The nonlocal dbar problem dressing method of Zakharov and Manakov (based on the approach of Ablowitz, Bar Yaacov, and Fokas to the IST for the KP equation) provides a candidate for a general expression of the local solutions to completely integrable systems. In this talk I will describe some recent results that give evidence that nonlocal dbar problem dressing method has the potential to provide a general local theory of real solutions to the KdV equation, and I will describe a framework to pursue analogous results for the KP equation. I will also discuss some numerical experiments based on this theory, and potential avenues for further asymptotic analysis.