Scattering in spacetime cones for the nonlinear Schrödinger equation

We consider the nonlinear Schrödinger equation on ${\mathbb{R}}^{3,1}$, with a class of `mixed’ nonlinearities that admit soliton solutions. We are interested in the long-time behaviour of solutions with finite, but possibly large, mass and/or energy. By a spacetime cone, we mean an open set in ${\mathbb{R}}_{x,t}^{3,1}$ that is invariant under dilation $(x,t)\mapsto (\alpha x,\alpha t)$ for all $\alpha >0$. Our main result is as follows: let $u(x,t)$ be the solution with initial condition in $H^2({\mathbb{R}}^3)$. Let $\mathcal{M},\mathcal{E}$ be the mass and energy of $u$, and let $\eta =\eta (\mathcal{M},\mathcal{E})>0$ be sufficiently small. Let $C_{\epsilon }$ be the complement of $N\ge 1$ of sufficiently thin spacetime cones $\{(x,t)\mid \mathrm{\mid }x-v_{j}t\mathrm{\mid }\le \epsilon t\}$, where $v_1,\dots ,v_N$ are fixed velocities and $\epsilon$ is sufficiently small. If the $L^{\mathrm{\infty }}$ norm of $u$ in $C_{\epsilon }$ is $\le \eta$, we prove that $u$ scatters in $C_{2\epsilon }\subset C_{\epsilon }$.