Scattering in spacetime cones for the nonlinear Schrödinger equation

Abstract

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 $\mathbb{R}^{3,1}_{x,t}$ that is invariant under dilation $(x, t) \mapsto (αx, αt)$ for all $α> 0$. Our main result is as follows: let $u(x, t)$ be the solution with initial condition in $H^2(\mathbb{R}^3)$. Let $\mathscr{M}, \mathscr{E}$ be the mass and energy of $u$, and let $η= η(\mathscr{M}, \mathscr{E}) > 0$ be sufficiently small. Let $C_ε$ be the complement of $N \geq 1$ of sufficiently thin spacetime cones $\{ (x, t) \mid |x-v_jt| \leq εt \}$, where $v_1, \dots, v_N$ are fixed velocities and $ε$ is sufficiently small. If the $L^\infty$ norm of $u$ in $C_ε$ is $\leq η$, we prove that $u$ scatters in $C_{2ε} \subset C_ε$.