The Nonlinear Schrödinger Equation and the Landau-Lifshitz Equation

Abstract

The following sections are devoted to the discussion of one of the most important properties of integrable systems, their gauge equivalence. As an example, we discuss two systems most frequently arising in condensed matter physics: the nonlinear Schrödinger equation and the Landau-Lifshitz equation. Let us establish the connection between them and show, via gauge symmetry (equivalence), one can generate solutions in the frame of a given system or find solutions to a given equation by the use of solutions of a different equation. We will show that a broad range of known symmetry transformations such as the Galilei, Bäcklund, etc., transformations are embedded into the gauge ones [12]. Writing the linear spectral problem in the form of the (A₀,A₁) pair allows a formulation in the language of gauge field theory and to apply the powerful apparatus of this theory. In particular, using algebraic-geometric concepts such as fibre spaces, and homogeneous and symmetric spaces, it is possible to give a consistent algebraic-geometric interpretation of the relations under consideration and even in ‘geometrizing’ the interaction.