The Soliton and Its History

Abstract

We review the history of the soliton, since the first recorded observation of the ‘great solitary wave’ by Russell in 1834, as a means of developing the mathematical properties of a large class of solvable nonlinear evolution equations. This class embraces, amongst others, the Korteweg-de Vries, sine-Gordon, and nonlinear Schrödinger equations. Solitary waves, solitons, Bäcklund transformations, conserved quantities and integrable evolution equations as completely integrable Hamiltonian systems are all introduced this way and form a basis for the more detailed discussions which follow in the remaining chapters. The differential geometry of one large class of nonlinear evolution equations is described. Some connections with nonlinear field theories and with solvable many-body problems are established. The short biography of John Scott Russell which forms much of the first section is continued as an appendix at the back of this volume.KeywordsSolitary WaveSoliton SolutionNonlinear Evolution EquationSolitary Wave SolutionIntegrable Hamiltonian SystemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.