## ABSTRACT

A method for solving the Cauchy problem for decaying initial data for integrable evolution equations in two spatial dimensions emerged in the early 1980s. This method is sometimes referred to as the ∂ method. Recall that the inverse spectral method for solving nonlinear evolution equations on the line is based on a matrix Riemann–Hilbert problem. There exists a large class of nonlinear evolution partial differential equations in one space variable which can be treated analytically. The most well-known integrable equations are nonlinear Schrodinger, the Korteweg. The book deVries and the sine-Gordon equations. Some integrable evolution equations in one space dimension possess particular solutions, which are localized in space and which retain their shape upon interaction with any other localized disturbance. Every integrable nonlinear evolution equation in one spatial dimension has several integrable versions in two spatial dimensions. There exist two types of localized coherent structures associated with integrable evolution equations in two spatial variables: the lumps and the dromions.