Global solutions for cubic quasilinear ultrahyperbolic Schr\"odinger flows

In recent work, two of the authors proposed a broad global well-posedness conjecture for cubic quasilinear dispersive equations in two space dimensions, which asserts that global well-posedness and scattering holds for small initial data in Sobolev spaces. As a first validation they proved the conjecture for quasilinear Schr\"odinger flows. In the present article we expand the reach of these ideas and methods to the case of quasilinear ultrahyperbolic Schr\"odinger flows, which is the first example with a nonconvex dispersion relation. The study of local well-posedness for this class of problems, in all dimensions, was initiated in pioneering work of Kenig-Ponce-Vega for localized initial data, and then continued by Marzuola-Metcalfe-Tataru (MMT) and Pineau-Taylor (PT) for initial data in Sobolev spaces in the elliptic and non-elliptic cases, respectively. Our results here mirror the earlier results in the elliptic case: (i) a new, potentially sharp local well-posedness result in low regularity Sobolev spaces, one derivative below MMT and just one-half derivative above scaling, (ii) a small data global well-posedness and scattering result at the same regularity level. One key novelty in this setting is the introduction of a new family of interaction Morawetz functionals which are suitable for obtaining bilinear estimates in the ultrahyperbolic setting. We remark that this method appears to be robust enough to potentially be of use in a large data regime when the metric is not a small perturbation of a Euclidean one.