The nonlinear Schr\"odinger equation with combined nonlinearities in 1D

We consider the one-dimensional nonlinear Schr\"odinger equation $$iu_t + u_{xx} + \mathcal{N}(u)u=0, \quad x,t \in \mathbb R,$$ with the nonlinearity term that is expressed as a sum of powers, possibly infinite: $$\mathcal{N}(u) = \sum d_k |u|^{\alpha_k}, \quad \alpha_k > 0.$$ We first investigate the local well-posedness of this equation for any positive powers of $\alpha_k$ in a certain weighted class of initial data, subset of $H^1 (\mathbb R)$. For that we use an approach of Cazenave-Naumkin [19], 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 behavior of such global solutions and prove scattering for data with the quadratic phase $e^{ib|x|^2}$ with sufficiently large positive $b$, in $H^1(\mathbb R)$. One of the advantages of considering an infinite sum in the nonlinearity term is being able to consider exponential nonlinearities, such as $e^{\gamma |u|^{k}} u$, 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 of combined nonlinearities, including the double nonlinearity and an exponential one, then investigate the behavior of solutions with positive or negative initial $b$ in a quadratic phase data. Furthermore, we also show that a ground state in the NLS equation with combined nonlinearities no longer provides a sharp threshold for global behavior such as scattering vs. finite time blow-up, instead the equation has a much richer dynamics.