We consider the one-dimensional nonlinear Schrödinger equation with the nonlinearity term that is expressed as a sum of powers, possibly infinite 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 pseudo-conformal 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 with sufficiently large positive power. In particular, we prove scattering of these solutions in H1(R). One of the advantages of considering an infinite sum in the nonlinearity term is being able to consider exponential nonlinearities, as well as sine or cosine nonlinearities, and obtain well-posedness in those cases, the first such result for most of those nonlinearities. To conclude, we show numerical simulations in the focusing case for various examples 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.
Nonlinear Schrödinger Equation with Combined Nonlinearities
Gia Azcoitia, Florida International UniversityAuthors: Gia Azcoitia, Alex David Roudriguez, Svetlana Roudenko, Oscar Riano, Hannah Wubben
2022 AWM Research Symposium