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Effects of nonlinear dynamics of solitary waves and wave modulations withinthe modular (also known as quadratically cubic) Korteweg - de Vries equationare studied analytically and numerically. Large wave events can occur in thecourse of interaction between solitons of different signs. Stable and unstable(finite-time-lived) breathers can be generated in inelastic collisions ofsolitons and from perturbations of two polarities. A nonlinear evolutionequation on long modulations of quasi-sinusoidal waves is derived, which is themodular or quadratically cubic nonlinear Schr\"odinger equation. Its solutionsin the form of envelope solitons describe breathers of the modular Korteweg -de Vries equation. The instability conditions are obtained from the linearstability analysis of periodic wave perturbations. Rogue-wave-type solutionsemerging due to the modulational instability in the modular Korteweg - de Vriesequation are simulated numerically. They exhibit similar wave amplification,but develop faster than in the Benjamin - Feir instability described by thecubic nonlinear Schr\"odinger equation.

Nonlinear waves, modulations and rogue waves in the modular Korteweg-de Vries equation