Elementary and binary Darboux transformations at rings

Abstract

For a given idempotent p and some element σ from a differential associative ring, we introduce gauge transformation λp + σ with the spectral parameter λ that leaves some linear operators form invariant. The explicit form of the σ is derived for the generalized Zakharov-Shabat problem. The maps that factorize Darboux transformations are referred to as elementary ones. Binary transformations that correspond to the iteration of elementary maps with the special choice of solutions of the direct and conjugate problems are introduced and widen the potential reductions set. We show an infinitesimal version of iterated transforms. The spectral operator and soliton theories applications are outlined. The nonabelian N-wave interaction equation modification with additional linear terms, its zero curvature representation, soliton solutions, and their stability with respect to infinitesimal deformations are studied.