Summary: We consider the following half wave Schrödinger equation, \[ (i \partial_t + \partial_x^2 - | D_y |) U = | U |^2 U \] 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 Szegő equation, which is similar to the existence result of modified wave operators on \(\mathbb{R}_x \times \mathbb{T}_y\) obtained by H. Xu [Math. Z. 286, No. 1–2, 443–489 (2017; Zbl 1367.35159)]. We then combine our modified wave operators result with a recent cascade result [“An inverse problem for Hankel operators and turbulent solutions of the cubic Szegő equation on the line”, J. Eur. Math. Soc. (to appear)] for the cubic Szegő equation by P. Gérard and A. Pushnitski to deduce that there exist solutions \(U\) to the half wave Schrödinger equation such that \(\| U ( t ) \|_{L_x^2 H_y^1}\) tends to infinity as \(\log t\) when \(t \to + \infty \). It indicates that the half wave Schrödinger 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.