  Traveling waves are ubiquitous in nature and control the speed of many
important dynamical processes, including chemical reactions, epidemic
outbreaks, and biological evolution. Despite their fundamental role in complex
systems, traveling waves remain elusive because they are often dominated by
rare fluctuations in the wave tip, which have defied any rigorous analysis so
far. Here, we show that by adjusting nonlinear model details, noisy traveling
waves can be solved exactly. The moment equations of these tuned models are
closed and have a simple analytical structure resembling the deterministic
approximation supplemented by a nonlocal cutoff term. The peculiar form of the
cutoff shapes the noisy edge of traveling waves and is critical for the correct
prediction of the wave speed and its fluctuations. Our approach is illustrated
and benchmarked using the example of fitness waves arising in simple models of
microbial evolution, which are highly sensitive to number fluctuations. We
demonstrate explicitly how these models can be tuned to account for finite
population sizes and determine how quickly populations adapt as a function of
population size and mutation rates. More generally, our method is shown to
apply to a broad class of models, in which number fluctuations are generated by
branching processes. Because of this versatility, the method of model tuning
may serve as a promising route toward unraveling universal properties of
complex discrete particle systems.
