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