Solitons are some remarkable solutions of certain nonlinear wave equations which behave in several ways like extended particles: they have a finite and localized energy, a characteristic velocity of propagation and a structural persistence which is maintained even when two solitons collide. Soliton waves propagating in a dispersive medium are the result of a balance between nonlinear effects and wave dispersion and therefore are only found in a very special class of nonlinear equations. Soliton waves were first found in some two-dimensional nonlinear differential equations in fluid dynamics such as the Korteweg–de Vries equation for shallow water waves. In the 1960s a method, known as the Inverse Scattering Method (ISM) was developed [111] to solve this equation in a systematic way and it was soon extended to other nonlinear equations such as the sine-Gordon or the nonlinear Schröodinger equations.

In the late 1970s the ISM was extended to general relativity to solve the Einstein equations in vacuum for spacetimes with metrics depending on two coordinates only or, more precisely, for spacetimes that admit an orthogonally transitive two-parameter group of isometries [23, 24, 206]. These metrics include quite different physical situations such as some cosmological, cylindrically symmetric, colliding plane waves, and stationary axisymmetric solutions. The ISM was also soon extended to solve the Einstein–Maxwell equations [4]. The ISM for the gravitational field is a solution-generating technique which allows us to generate new solutions given a background or seed solution.