Gauge covariant theory of the generating operator. I

Abstract

A gauge covariant formulation of the generating operator (Λ-operator) theory for the Zakharov-Shabat system is proposed. The operator\(\tilde \Lambda \), corresponding to the gauge equivalent system in the pole gauge is explicitly calculated. Thus the unified approach to the nonlinear Schrödinger-type equations based on Λ is automatically reformulated with the help of\(\tilde \Lambda \) for the Heisenberg ferromagnet-type equations. Consequently, it is established that the conserved densities for the Heisenberg-ferromagnet-type equations are polynomial inS(x) and itsx-derivatives. Special attention is paid to the interrelation between the hierarchies of symplectic structures corresponding to the above mentioned families of gauge-equivalent equations. It is shown that the geometrical properties of the conjugated operator Λ* are gaugeindependent.