Nonlinear dynamics of quadratically cubic systems

We propose a modified form of the well-known nonlinear dynamic equations with quadratic relations used to model a cubic nonlinearity. We show that such quadratically cubic equations sometimes allow exact solutions and sometimes make the original problem easier to analyze qualitatively. Occasionally, exact solutions provide a useful tool for studying new phenomena. Examples considered include nonlinear ordinary differential equations and Hopf, Burgers, Korteweg – de Vries, and nonlinear Schrödinger partial differential equations. Some problems are solved exactly in the space–time and spectral representations. Unsolved problems potentially solvable by the proposed approach are listed.