On generalized Loewner systems: Novel integrable equations in 2+1 dimensions

A reinterpretation and generalization of a class of infinitesimal Bäcklund transformations originally introduced in a gas-dynamics context by Loewner in 1952 leads to a linear representation for a novel class of 2+1-dimensional nonlinear equations. The latter may be parametrized in terms of a triad of eigenfunctions. Moreover, like the well-known integrable Davey–Stewartson and Nizhnik–Novikov–Veselov equations, the nonlinear systems typically contain the two spatial variables on an equal footing. The ∂̄-dressing method is outlined for these generalized Loewner systems. It is noted that basic reductions lead to 2+1-dimensional versions of the principal chiral-fields model, Toda lattice system, and notably the classical sine–Gordon equation. Loewner systems on Grassmannian, the projective CPn and RPn manifolds, are considered. In particular, the 2+1-dimensional integrable sine–Gordon system in which x and y occur in a symmetric manner arises naturally in this context. A gauge-equivalent system likewise emerges out of a special reduction of a 2+1-dimensional Toda lattice scheme constructed here via the Loewner formalism.