The nonlinear Schrödinger equation with combined nonlinearities in 1D

Abstract We consider the one-dimensional nonlinear Schrödinger equation i u t + u x x + N ( u ) u = 0 , x , t ∈ R , with the nonlinearity term that is expressed as a sum of powers, possibly infinite: N ( u ) = ∑ d k | u | α k , α k > 0. We first investigate the local well-posedness of this equation for any positive powers of α k in a certain weighted class of initial data, subset of H 1 ( R ) . For that we use an approach of Cazenave–Naumkin (Cazenave and Naumkin 2016 Commun. Contemp. Math. 19 1650038), thus, avoiding any Strichartz estimates. Then, using the pseudo-conformal transformation, we extend the local result to the global one for the initial data with a quadratic phase. Furthermore, we investigate the asymptotic behaviour of such global solutions and prove scattering for data with the quadratic phase e i b | x | 2 with sufficiently large positive b , in H 1 ( R ) . One of the advantages of considering an infinite sum in the nonlinearity term is being able to consider exponential nonlinearities, such as e γ | u | k u , 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 for various examples