Canonical and Noncanonical Recursion Operators in Multidimensions

The hierarchies of evolution equations associated with the spectral operators ?_x?_y ? R?_y ? Q and ?_x?_y ? Q in the plane are considered. In both cases a recursion operator Ф₁₂, which is nonlocal and generates the hierarchy, is obtained. It is shown that only in the first case does the recursion operator satisfy the canonical geometrical scheme in 2 + 1 dimensions proposed by Fokas and Santini. The general procedure proposed allows one to derive, at the same time, the evolution equations associated with a given linear spectral problem and their Backlund transformations (if they exist), with no need to verify by long and tedious computations the algebraic properties of Ф₁₂. Two equations in the first hierarchy can be considered as two different integrable generalizations in the plane of the dispersive long wave equation. All equations in this hierarchy are shown to be both a dimensional reduction of bi-Hamiltonian n × n matrix evolution equations in multidimensions and a generalization in the plane of bi-Hamiltonian n × n matrix evolution equations on the line.