ABSTRACT
This chapter introduces Scott Russell’s observation and points out how the Korteweg–de Vries (KdV) solitary wave represents the Scott Russell phenomenon. The KdV equation appeared as a sound basic formulation of the Scott Russell phenomenon when the Dutch physicists Korteweg and de Vries derived it from first principles and showed explicitly that it admits amplitude-dependent cnoidal nonlinear dispersive wave solutions, a limiting form of which is the solitary wave. Starting from the basic principles of hydrodynamics and considering unidirectional wave propagation in a long but shallow channel, they deduced the celebrated wave equation responsible for the phenomenon, which now goes by their names. The KdV equation is a simple nonlinear dispersive wave equation. Norman Zabusky and Martin Kruskal recalled that the KdV equation admits a solitary wave solution, which has a distinct nonlinear character. The chapter discusses the Hamiltonian structure and complete integrability aspects of the KdV equation.
