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

We consider the following half wave Schrodinger equation, (iat+ax2- |Dy|)U= |U|2U on the plane RxxRy. We prove the existence of modified wave operators between small decaying solutions to this equation and small decaying solutions to the non chiral cubic Szeg6 equation, which is similar to the existence result of modified wave operators on Rx x Ty obtained by H. Xu [20]. We then combine our modified wave operators result with a recent cascade result [11] for the cubic Szeg6 equation by P. Gerard and A. Pushnitski to deduce that there exist solutions U to the half wave Schrodinger equation such that fU(t)fL2xHy1 tends to infinity as logt when t-+ +oo. It indicates that the half wave Schrodinger 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.(c) 2023 Elsevier Inc. All rights reserved.

Automated summary

This article investigates the relationship between the half wave Schrodinger equation and the non-chiral cubic Szeg6 equation, and proves the existence of modified wave operators between them. Meanwhile, by combining with other research results, it deduces the characteristic of the global solutions for the half wave Schrodinger equation.