D-bar Methods in Direct and Inverse Problems

The Inverse Scattering Transform is considered one of the most important breakthroughs of mathematical physics in the 20th century. Its origins lie in the explicit solution of the Korteweg-de Vries equation for shallow water waves, but the resulting approach can be applied to other nonlinear partial differential equations as well. The method involves transforming a nonlinear PDE (in the physical space) into a linear spectral problem (in the frequency domain). In the 1980s, Beals and Coifman considered framing the inverse scattering method as the inversion of a D-bar equation, thus allowing generalizations to multiple dimensions and the study of additional classes of nonlinear PDEs. D-bar methods have proved very useful in evolution equations as well as inverse problems by allowing the rigorous study of the full nonlinear problems in the setting of constructive proofs, thus providing a natural bridge towards numerical implementations. In this talk, we explore the D-bar method for an evolution equation (the Davey-Stewartson II defocusing equation) and the inverse problem of Electrical Impedance Tomography (via the Schrödinger equation). Numerical evolutions/reconstructions are presented using simulated data.