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

Conformal duality of the nonlinear Schr\"odinger equation: Theory and applications to parameter estimation