We construct pure two-bubbles for the energy-critical focusing nonlinear Schrödinger equation in space dimension . 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 . One of the bubbles develops at scale , whereas the length scale of the other converges to at rate . The phases of the two bubbles form the right angle.
"Construction of two-bubble solutions for the energy-critical NLS." Anal. PDE 10 (8) 1923 - 1959, 2017. https://doi.org/10.2140/apde.2017.10.1923