Strongly interacting blow up bubbles for the mass critical NLS

TL;DR

This paper constructs novel multi-solitary wave solutions for the mass critical 2D NLS, demonstrating complex blow-up behaviors with strong interactions, including infinite and finite time blow-up with rates exceeding classical predictions.

Contribution

It introduces a new class of solutions with multiple bubbles interacting strongly, leading to unprecedented blow-up rates in the mass critical NLS.

Findings

Existence of solutions with multiple bubbles at polygon vertices

Solutions blow up in infinite time with logarithmic rate

First example of finite-time blow-up exceeding pseudo-conformal rate

Abstract

We construct a new class of multi-solitary wave solutions for the mass critical two dimensional nonlinear Schrodinger equation (NLS). Given any integer K>1, there exists a global (for positive time) solution of (NLS) that decomposes asymptotically into a sum of solitary waves centered at the vertices of a K-sided regular polygon and concentrating at a logarithmic rate in large time. This solution blows up in infinite time with logarithmic rate. Using the pseudo-conformal transform, this yields the first example of solution blowing up in finite time with a rate strictly above the pseudo-conformal one. Such solution concentrates K bubbles at a point. These special behaviors are due to strong interactions between the waves, in contrast with previous works on multi-solitary waves of (NLS) where interactions do not affect the blow up rate.