  We show the relationship between the strongly non-linear limit (also termed
the dispersionless or the Whitham limit) of the macroscopic fluctuation theory
of certain statistical models and the inverse scattering method. We show that
in the strongly non-linear limit the inverse scattering problem can be solved
using the steepest descent method of the associated Riemann--Hilbert problem.
The importance of establishing this connection, is that the equations in the
strongly non-linear limit can often be solved exactly by simple means, the
connection then provides a limit in which one can solve the inverse scattering
problem, thus aiding potentially the exact solution of a particular large
deviation problem.
