Introduction to the Hirota Direct Method

The primary subject matter of the report is the Hirota Direct Method, and the primary goal of the report is to describe and derive the method in detail, and then use it to produce analytic soliton solutions to the Boussinesq equation and the Korteweg-de Vries (KdV) equation. Our hope is that the report may also serve as an introduction to soliton theory at an undergraduate level. The report follows the structure of first introducing Hirota's bi-linear operator and giving an account of its relevant properties. The properties of the operator are then used to find soliton solutions for differential equations that can be expressed in a "bilinear" form. Thereafter, a set of methods for finding the bilinear form of a more general non-linear differential equation are presented. Finally, we apply the tools to the Boussinesq and KdV equations respectively to derive their soliton solutions