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.