Global Well-Posedness of Quadratic and Subquadratic Half Wave Schrödinger Equations

In this pages, we consider the p order nonlinear half wave Schrödinger equations aligned ( i ∂ ₜ+∂ ₓ²-| Dy| ) u=± |u|ᵖ⁻¹ u aligned i ∂ t + ∂ x 2 - D y u = ± | u | p - 1 u on the plane R² R 2 with 1<p≤ 2 1 < p ≤ 2 . We prove the global well-posedness of this equation in Lₓ² Hyˢ(R²) ∩ Hₓ¹ Ly²(R²) L x 2 H y s ( R 2 ) ∩ H x 1 L y 2 ( R 2 ) ( 1/2≤ s ≤ 1 1 2 ≤ s ≤ 1 ), which is the first global well-posedness result of nonlinear half wave Schrödinger equations. With the global well-posedness in the energy space for the focusing equation and the study on the solitary wave in Bahri et al. (Commun Contemp Math 23(05), 2020), we complete the proof of the stability of the set of ground states. Moreover, we consider the half wave Schrödinger equations on Rₓ× Ty R x × T y , which can also be called the wave guide Schrödinger equations on Rₓ× Ty R x × T y