  We study a deformation of the nonlinear Schr\"odinger equation recently
derived in the context of deformation of hierarchies of integrable systems.
This systematic method also led to known integrable equations such as the
Camassa-Holm equation. Although this new equation has not been shown to be
completely integrable, its solitary wave solutions exhibit typical soliton
behaviour, including near elastic collisions. We will first focus on standing
wave solutions, which can be smooth or peaked, then, with the help of numerical
simulations, we will study solitary waves, their interactions and finally rogue
waves in the modulational instability regime. Interestingly the structure of
the solution during the collision of solitary waves or during the rogue wave
events are sharper and have larger amplitudes than in the classical NLS
equation.
