Nonlinear Dynamics on the Plane and Integrable Hierarchies of   Infinitesimal Deformations
  A class of nonlinear problems on the plane, described by nonlinear inhomogeneous $\bar{\partial}$-equations, is considered. It is shown that the corresponding dynamics, generated by deformations of inhomogeneous terms (sources) is described by Hamilton-Jacobi type equations associated with hierarchies of dispersionless integrable systems. These hierarchies are constructed by applying the quasiclassical $\bar{\partial}$-dressing method. 
B. Konopelchenko and L. Martinez Alonso