The direct linearizing transform for three-dimensional lattice equations (Q1092241)

In this extended abstract the author presents the direct linearizing transform (DLT) for a general class of nonlinear partial difference equations on a three-dimensional lattice. The main idea is to start directly from a linear integral equation in the spectral space, with integrations over an arbitrary measure and contour. Not only does one expect to capture in this way a large class of solutions of nonlinear equations, but it turns out also that the DLT gives a very clear insight into the various relations between nonlinear equations, such as Miura transformations, Bäcklund transformations and the construction of integrable discretizations of partial differential equations. The DLT for three-dimensional lattice equations as presented here, unifies in a way all former results.