Hirota's bilinear method and soliton solutions

In this lecture we will flrst discuss integrability in general, its meaning and signiflcance, and then make some general observations about solitons. We will then introduce Hirota’s bilinear method, which is particularly useful in constructing multisoliton solutions for integrable nonlinear evolution equations. 1 Why is integrability important? In very general terms integrability means regularity in time evolution. This is due to the existence of many conserved quantities. Most dynamical systems have some conserved quantities, such as energy or total momentum, but integrable systems have more of them. How many is enough depends on the system. For example, a Hamiltonian system of N coordinates and N momenta is said to be Liouville integrable if it has N conserved quantities, which furthermore must be su‐ciently regular (analytic) and mutually commuting under the Poisson bracket. In principle the system can then be solved by quadratures. The existence of such a large quantity of conserved quantities is a special requirement for the