Approximate Peregrine Solitons in Dispersive Nonlinear Wave Equations

The purpose of this short note is to explain how the existing results on the validity of the NLS approximation can be extended from Sobolev spaces $H^s(\mathbb{R})$ to the spaces of functions $u = v + w$ where $v \in H_{{per}}^s$ and $w \in H^s(\mathbb{R})$. This allows us to use the Peregrine solution of the NLS equation to find freak or rogue wave dynamics in more complicated systems.