ABSTRACT
One of the first nonlinear ODEs considered analytically by mathematicians was the Riccati equation. The study of the nonlinearity of the Riccati equation reveals that it can be converted to a system of linear ODEs. This chapter discusses the soliton solution. The term soliton was proposed by Zabusky and Kruskal in 1965 to describe a localized solitary wave with a permanent form. This wave does not disperse with distance and does not amplify with distance. It violates the normal principle of superposition. This chapter discusses the application of a shock wave to traffic flow problems. The nonlinear growth of wave cancels the dispersive decay of the wave resulting in a stable non-decaying shape-preserving soliton solution. In 1968, Zakharov was the first to show that deep water waves are governed by the nonlinear Schrödinger equation, and derived the breather-type solution through the Benjamin-Feir instability.
