  We consider the following half wave Schr{\"o}dinger equation,$$\left(i
\partial_{t}+\partial_{x }^2-\left|D_{y}\right|\right) U=|U|^{2} U$$on the
plane $\mathbb{R}_x \times \mathbb{R}_y$. We prove the existence of modified
wave operators between small decaying solutions to this equation and small
decaying solutions to the non chiral cubic Szeg\H o equation, which is similar
to the the existence result of modified wave operators on $\mathbb{R}_x \times
\mathbb{T}_y$ obtained by H. Xu [16]. We then combine our modified wave
operators result with a recent cascade result [7] for the cubic Szeg\H o
equation by P. G{\'e}rard and A. Pushnitski to deduce that there exists
solutions $U$ to the half wave Schr{\"o}dinger equation such that
$\|U(t)\|_{L_{x}^{2} H_{y}^{1}}$ tends to infinity as $\log t$ when $t
\rightarrow +\infty$. It indicates that the half wave Schr{\"o}dinger equation
on the plane is one of the very few dispersive equations admitting global
solutions with small and smooth data such that the $H^s$ norms are going to
infinity as $t$ tends to infinity.
