From nonlinear Burgers and Korteweg–de Vries soliton equations via Riccati to linear Rosen–Morse and free particle Schrödinger equations

Abstract

It is shown that a nonlinear reformulation of time-dependent and time-independent quantum mechanics in terms of Riccati equations not only provides additional information about the physical system, but also allows for formal comparison with other nonlinear theories. This is demonstrated for the nonlinear Burgers and Korteweg–de Vries equations with soliton solutions. As Riccati equations can be linearized to corresponding Schrödinger equations, this also applies to the Riccati equations that can be obtained by integrating the nonlinear soliton equations, resulting in a time-independent Schrödinger equation with Rosen–Morse potential and its supersymmetric partner. Because both soliton equations lead to the same Riccati equation, relations between the Burgers and Korteweg–de Vries equations can be established. Finally, a connection with the inverse scattering method is mentioned.