Solutions of the Kadomtsev-Petviashvili equation associated to higher rank commuting ordinary differential operators

This thesis uses explicit formulae for higher rank commuting ordinary differential operators to find explicit solutions of the Kadomtsev-Petviashvili equation. The method for doing this uses the formulation of the Kadomtsev-Petviashvili equation as a compatibility condition for a formal pseudo-differential operator.;A by-product of the method is a generalization of previously known rank two commuting differential operators to higher genus and also some higher degree rank three commuting differential operators.