Construction of two-bubble blow-up solutions for the mass-critical gKdV equations

For the mass-critical generalized Korteweg-de Vries equation, $$\partial_{t}u+\partial_{x}\left( \partial_{x}^{2}u+u^{5}\right)=0,\quad (t,x)\in [0,\infty)\times \mathbb{R}.$$ We prove the existence of a global solution that blows up in infinite time and approaches the sum of two decoupled bubbles with opposite signs. The proof is inspired by the techniques developed for the two-dimensional mass-critical NLS equation in a similar context by Martel-Rapha\"el [37]. The main difficulty originates from the fact that the unstable directions related to scaling are excited by the nonlinear interactions. To overcome this difficulty, a refined approximate solution that involves some non-localized profiles is needed. In particular, a sharp understanding for the interactions between solitons and such profiles is also required.