On the beta distribution, the nonlinear Fourier transform and a combinatorial problem

Abstract

The paper describes some probabilistic and combinatorial aspects of the nonlinear Fourier transform associated with the AKNS-ZS problems. One of the two main results shows that the volumes of a family of polytopes that appear in a power expansion of the nonlinear Fourier transform are distributed according to the beta probability distribution. We establish this result by studying an Euler-type discretisation of the nonlinear Fourier transform. This approach provides another main result. We discover a novel discrete probability distribution approximating the beta distribution with integral shape parameters. For specific choices of the shape parameters, our new discrete distribution has a natural combinatorial interpretation; the numbers of alternating ordered partitions of an integer into distinct parts are essentially distributed according to our new discrete distribution. The generating function for these numbers can be easily expressed using the nonlinear Fourier transform of the constant function.