Construction of two-bubble solutions for the energy-critical NLS

TL;DR

This paper constructs special two-bubble solutions for the energy-critical focusing nonlinear Schrödinger equation in high dimensions, revealing detailed asymptotic behavior and phase relations of the bubbles.

Contribution

It introduces a method to construct pure two-bubble solutions with specific scale and phase properties in dimensions $N \,\geq\, 7$, advancing understanding of multi-bubble dynamics.

Findings

Two-bubble solutions are constructed with one bubble at fixed scale and the other shrinking to zero.

The scale ratio of the bubbles converges to zero as time approaches infinity.

The phases of the bubbles form a right angle, indicating a specific geometric configuration.

Abstract

We construct pure two-bubbles for the energy-critical focusing nonlinear Schr\"odinger equation in space dimension $N \geq 7$. The constructed solution is global in (at least) one time direction and approaches a superposition of two stationary states both centered at the origin, with the ratio of their length scales converging to $0$. One of the bubbles develops at scale $1$, whereas the length scale of the other converges to $0$ at rate $|t|^{-\frac{2}{N-6}}$. The phases of the two bubbles form the right angle.