Existence of modified wave operators and infinite cascade result for a half wave Schrödinger equation on the plane

Abstract

We consider the following half wave Schrödinger equation,(i∂t+∂x2−|Dy|)U=|U|2U on the plane Rx×Ry. 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 Rx×Ty obtained by H. Xu [20]. We then combine our modified wave operators result with a recent cascade result [11] 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)‖Lx2Hy1 tends to infinity as log⁡t when t→+∞. 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 Hs norms are going to infinity as t tends to infinity.