Until the beginning of this decade the number of significant exactly soluble problems in physics was limited to a very few: the classical or quanti sed harmonic oscillator, the linearised many-body problem, the quanti sed hydrogen atom, Newton's solution of the planetary orbit problem, Onsager's solution of the two-dimensional Ising problem, almost exhaust the list. Now the situation is quite different. We have a large number of exactly soluble nonlinear systems of physical significance and the number of these is growing steadily. Recent examples include a limited solution of Ein- stein's field equations!, an apparently exact solution of the quantised sine-Gordon 2 equation u - U = sin(u) which establishes connections with the Ising models, xx tt 3 and a solution of the equations of motion of a rigid body in g dimensions . This book is concerned with problems such as these. But more specifically it is concerned with solitons. These mathematical objects are exact, analytical, solutions of nonlinear wave or evolution equations like the sine-Gordon equation (s-G) or the Korteweg-de Vries equation (KdV) u ] 6uu + u = O. The discovery by Gardner, t x xxx .