Recursion operators and bi-Hamiltonian structures in multidimensions. I
In a previous paper [Stud. Appl. Math. 75, 179-185 (1986; Zbl 0613.35073)] the authors constructed surprisingly a recursion operator for a famous \(2+1\) (i.e., 2 spatially and one temporally) dimensional hierarchy of nonlinear evolution equations. In the present paper under review the authors develop a general theory for the algebraic structure of recursion operators in \(2+1\) dimensions. A number of important notions, especially the hereditary symmetry and the Hamiltonian pair, that were developed thus far for \(1+1\) dimensions, are successfully extended to the \(2+1\) dimensions. As illustrative examples the authors study the hierarchies related to the following two linear eigenvalue problems: \[ w_{xx}+q(x,y)w+\alpha w_ y=0, \] where \(\alpha\) is a constant, and \(W_ x=JW_ x+QW\) where \(J=diag(\alpha,-\alpha),\) Q is a \(2\times 2\) off-diagonal matrix containing the potentials \(q_ 1(x,y)\) and \(q_ 2(x,y)\).
hierarchy of nonlinear evolution equations
scientific article; zbMATH DE number 4103620
scientific article