Among the most prominent outstanding problems in physics is to develop a more complete understanding of gravity. General relativity, while very successful and predictive, ultimately breaks down at high energy densities where it is proposed that quantum mechanical effects begin to dominate. One theory that seeks to solve this incompatibility in a cosmologically relevant way is loop quantum cosmology (LQC). Inherent in LQC is the discretization of space, which leads to difference equations as the descriptive language of very early universe evolution. This thesis examines the use of the Inverse Scattering Transform (IST) as a method to solve nonlinear difference equations expected to arise from loop quantum cosmology. We also discuss the progress made thus far in generating the relevant nonlinear difference equation.
