  One outstanding problem in the physics of glassy solids is understanding the
statistics and properties of the low-energy excitations that stem from the
disorder that characterizes these systems' microstructure. In this work we
introduce a family of algebraic equations whose solutions represent collective
displacement directions (modes) in the multi-dimensional configuration space of
a structural glass. We explain why solutions of the algebraic equations, coined
nonlinear glassy modes, are quasi-localized low-energy excitations. We present
an iterative method to solve the algebraic equations, and use it to study the
energetic and structural properties of a selected subset of their solutions
constructed by starting from a normal mode analysis of the potential energy of
a model glass. Our key result is that the structure and energies associated
with harmonic glassy vibrational modes and their nonlinear counterparts
converge in the limit of very low frequencies. As nonlinear modes never suffer
hybridizations, our result implies that the presented theoretical framework
constitutes a robust alternative definition of `soft glassy modes' in the
thermodynamic limit, in which Goldstone modes overwhelm and destroy the
identity of low-frequency harmonic glassy modes.
