  The nonlinear Schr\"odinger equation (NLSE) is a rich and versatile model,
which in one spatial dimension has stationary solutions similar to those of the
linear Schr\"odinger equation as well as more exotic solutions such as solitary
waves and quantum droplets. Here we present the unified theory of the NLSE,
showing that all stationary solutions of the local one-dimensional
cubic-quintic NLSE can be classified according to a single number called the
cross-ratio. Any two solutions with the same cross-ratio can be converted into
one another using a conformal transformation, and the same also holds true for
traveling wave solutions. Further, we introduce an optimization afterburner
that relies on this conformal symmetry to substantially improve NLSE parameter
estimation from noisy empirical data. The new method therefore should have far
reaching practical applications for nonlinear physical systems.
