A linearizing transformation for the Korteweg–de Vries equation; generalizations to higher-dimensional nonlinear partial differential equations

It is shown that the Korteweg–de Vries (KdV) equation can be transformed into an ordinary linear partial differential equation in the wave number domain. Explicit solutions of the KdV equation can be obtained by subsequently solving this linear differential equation and by applying a cascade of (nonlinear) transformations to the solution of the linear differential equation. It is also shown that similar concepts apply to the nonlinear Schrödinger equation. The role of symmetry is discussed. Finally, the procedure which is followed in the one-dimensional cases is successfully applied to find special solutions of higher-dimensional nonlinear partial differential equations.