Geometric theory for multi-bump, self-similar, blowup solutions of the cubic nonlinear Schrödinger equation

Abstract

We establish the existence and local uniqueness of two classes of multi-bump, self-similar, blowup solutions for the cubic nonlinear Schrödinger equation close to the critical dimension d = 2. Our results for one class of orbits build on the earlier discovery of these orbits via numerical simulation and via asymptotic analysis, providing a proof of their existence. The second class of multi-bump orbits is new. These multi-bump orbits, many of which are thought to be unstable, appear to serve as guides for how different types of initially non monotone data might blow up.These self-similar solutions are governed by a nonlinear, nonautonomous ordinary differential equation (ODE); and, when linearized, this ODE exhibits hyperbolic behaviour near the origin and elliptic behaviour asymptotically. In between, the behaviour changes type; this region is called the midrange. For the solutions of the full ODE that we construct, all but one of the bumps—the exception being the central bump at the origin—lie in the midrange. The main steps in the proof involve (i) tracking a pair of manifolds of solutions of the governing ODE that satisfy the conditions at the origin and the asymptotic conditions, respectively, to a common point in the midrange, and (ii) showing that these intersect transversally. Geometric singular perturbation theory, adiabatic Melnikov theory, and the exchange lemma are used to analyse the dynamics in the midrange.