Backlund Transformations and Three-dimensional Lati'ice Equations

1. The investigation of nonlinear evolution equations in multi-dimensional space-time has received much interest recently [ 1-7] . Although a lot of progress has been made in extending the inverse-scattering transform to higher-dimensional equations,such as the Kadomtsev-Petviashvili (KP) equation [1 ], some of the essential features are not completely understood. In order to obtain more insight in the structure of higher.dimensional integrable systems it may be useful to start from higher-dimensional partial difference equations (PDFEs) and their exact linearizations. The reason is that PDFEs with enough free parameters contain the complete information on the system, not only on the lattice, but also in the continuum. In fact, taking appropriate continuum limits one can derive integrable PDEs and their exact linearizations associated with various dispersions. For a two-dimensional lattice we have shown that the (algebraic) identities for the commutativity of B~cklund transformations (BTs) can be interpreted as integrable PDFEs [8]. In this letter we show that the same holds true for the three-dimensional case. In order to show this we study a linear integral equation which contains an integration over an arbitrary two-dimensional hypersurface in terms of two complex variables with an arbitrary measure depending on these two complex variables. In this case BTs are generated by nonlocal transformations of the measure, and for each BT one can derive linear relations for the wavefunctions of the problem. The compatibility of three of these relations under three different BTs leads to an algebraic indentity which can be interpreted as a three-dimensional lattice equation. By considering various continuum limits we derive the two-dimensional Todd equation and the KP equation. Finally a reduction to equations of lower dimensionality is discussed.