Geometry of generalised nonlinear Schrodinger and Heisenberg ferromagnetic spin equations with linearly x-dependent coefficients

Abstract

The integrable generalised nonlinear Schrodinger equation with linearly x-dependent coefficients is shown to be equivalent to the equation of motion of a generalised Heisenberg ferromagnet in the continuum limit. This is represented by the motion of a nonlinear string thereby clarifying its geometric structure. An (L', Â) pair is constructed for this string and the eigenvalues have a simple time evolution. Although these flows are not isospectral they all satisfy the vanishing curvature condition Θ≡dΩ-Ω∧Ω=0.