Please join us remotely for a seminar by FIU graduate student Alex David Rodriquez.

Rodriguez considers the one-dimensional nonlinear Schrödinger equation iut + uxx + N (u) = 0, x, t ∈ R, with the nonlinearity term that is expressed as a sum of powers, possibly infinite: N (u) = Xcn|u| αn u, αn > 0. The combined nonlinearities appear in various physical applications such as chemical super fluidity, or the description of elementary particles such as bosons and defectons, or other subatomic structures, and in approximations of anisotropic media. We first investigate the local well-posedness of this equation for any positive powers of α in a certain weighted class of initial data, subset of H1 (R). Then, using the pseudoconformal transformation, we extend the local result to the global well-posedness. Furthermore, we investigate the asymptotic behavior of global solutions, those that have initial data with a quadratic phase e ib|x| 2 with sufficiently large positive b, in particular, we prove scattering of these solutions in H1 (R). One of the advantages of considering an infinite sum in the nonlinear term is the investigation of an exponential nonlinearity e α|u|u and its well-posedness in that case, the first such result. To conclude, we show numerical simulations in the focusing case for various cases of combined nonlinearities, including the exponential one, and investigate a threshold behavior for the global versus finite time existing solutions, which extends our theoretical results. This talk is based on the joint work with Gia Azcotia, Oscar Riano, Svetlana Roudenko and Hannah Wubben, which was initiated during the Research Experiences for Undergraduates program “AMRPU @ FIU” in Summer 2021.

Hosted by Hakima Bessiah.