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

Whitham Approach to Certain Large Fluctuation Problems in Statistical Mechanics