Existence of modified wave operators and infinite cascade result for a half wave Schr{ö}dinger equation on the plane

We consider the following half wave Schr{ö}dinger equation,$$(i{\mathrm{\partial }}_t+{\mathrm{\partial }}_x^2-\mid D_y\mid )U=\mathrm{\mid }U{\mathrm{\mid }}^{2}U$$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{é}rard and A. Pushnitski to deduce that there exists solutions $U$ to the half wave Schr{ö}dinger equation such that $\mathrm{\parallel }U(t){\mathrm{\parallel }}_{L_x^2{H}_y^1}$ tends to infinity as $\log t$ when $t\to +\mathrm{\infty }$. It indicates that the half wave Schr{ö}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.