The second half of the twentieth century saw a resurgence in the study of classical physics. Scientists began paying particular attention to the effects caused by the nonlinearity in dynamical equations. This nonlinearity was found to have two interesting manifestations of opposite nature: chaos, that is the apparent randomness in the behaviour of perfectly deterministic systems, and solitons, that is localized, stable moving objects that scattered elastically. Both of these topics have now been developed into paradigms, with solid mathematical background and with a wide range of physical observations and concrete applications.

This book is concerned with a particular method used in the study of solitons. There are many ways of studying the integrable nonlinear evolution equations that have soliton solutions, each method having its own assumptions and areas of applicability. For example, the inverse scattering transform (IST) can be used to solve initial value problems, but it uses powerful analytical methods and therefore makes strong assumptions about the nonlinear equations. On the other hand, one can find a travelling wave solution to almost all equations by a simple substitution which reduces the equation to an ordinary differential equation. Between these two extremes lies Hirota's direct method. Although the transformation was, at its heart, inspired by the IST, Hirota's method does not need the same mathematical assumption and, as a consequence, the method is applicable to a wider class of equations than the IST.