A brief historical introduction to solitons and the inverse scattering transform–a vision of Scott Russell

Abstract

‘...the great primary waves of translation cross each other without change of any kind in the same manner as the small oscillations produced on the surface of a pool by a falling stone.’ SCOTT RUSSELL ‘...as Sir Cyril Hinshelwood has observed... fluid dynamicists were divided into hydraulic engineers who observed things that could not be explained and mathematicians who explained things that could not be observed.’ JAMES LIGHTHILL ‘If you wish to foresee the future of mathematics our proper course is to study the history and present condition of the science.’ HENRI POINCARÉ This paper deals with a brief introduction to major remarkable discoveries of the soliton and the inverse scattering transform in the 1960s. The discovery of the soliton (or the solitary waves) began with the famous physical experiments of the Scottish Engineer and Naval Architect John Scott Russell in the Glasgow–Edinburgh Canal in 1834. The main objective of this paper is to introduce Scott Russell's vision to the reader. This was followed by the famous mathematical derivation of the Korteweg-de Vries (KdV) equation in 1895 for the propagation of the great solitary wave of Scott Russell in one direction on the free surface of water in a shallow canal. In 1965, Norman Zabusky and Martin Kruskal discovered the existence of solitons and the interaction of solitons from their computer experiments. In 1967, C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura developed an ingenious method for finding the exact solution of the KdV equation. This paper is also concerned with several fundamental nonlinear partial differential equations including the Korteweg and de Vries (KdV) equations, the nonlinear Schrödinger (NLS) equation, the Sine-Gordon (SG) equation and the Toda lattice equation. Thus, the soliton and the inverse scattering transform are now regarded as the major remarkable discoveries in mathematical sciences of the second half of the twentieth century. A brief comment is made on the discovery of compacton in the 1990s and physical interaction of two or more compactons. A compacton is a soliton with a finite compact support with no oscillatory trail. Interestingly, compactons can describe the intrinsic localized modes in anharmonic crystals. Concluding remarks are made on the recent developments of nonlinear wave phenomena and their applications to a wide variety of problems.