Solution of the Cauchy problem for a continuous limit of the Toda lattice and its superextension (Q1177124)

The authors construct a supersymmetric equation associated with a continuum limit of the classical superalgebra sl\((n/n+1)\). This equation can be thought of as a superextension of a continuous limit of the Toda lattice with fixed endpoints or, alternatively, as a supersymmetric version of the so-called heavenly equation: \[ (\partial^ 2/\partial z_ +\partial z_ -)\rho = (\partial^ 2/\partial \tau^ 2)\exp \rho. \] This equation has many applications in physics. In particular, it describes the real self-dual Einstein spaces with one rotational Killing vector. Using a formal argument, the authors also give a solution of the Cauchy problem for the continuous limit of the Toda lattice and for its superextension.