Construction of multi-bubble solutions for the critical gKdV equation

Abstract

We prove the existence of solutions of the mass critical generalized Korteweg-de Vries equation ${\partial }_{t}u+{\partial }_x({\partial }_{xx}u+u^5)=0$ containing an arbitrary number $K\ge 2$ of blow up bubbles, for any choice of sign and scaling parameters: for any ${\ell }_1>{\ell }_2>\cdots >{\ell }_K>0$ and ${\epsilon }_1,\dots ,{\epsilon }_K\in \{\pm 1\}$, there exists an $H^1$ solution $u$ of the equation such that \[ u(t) - \sum_{k=1}^K \frac {ε_k}{λ_k^\frac12(t)} Q\left( \frac {\cdot - x_k(t)}{λ_k(t)} \right) \longrightarrow 0 \quad\mbox{ in }\ H^1 \mbox{ as }\ t\downarrow 0, \] with ${\lambda }_k(t)\sim {\ell }_{k}t$ and $x_k(t)\sim -{\ell }_k^{-2}{t}^{-1}$ as $t\downarrow 0$. The construction uses and extends techniques developed mainly by Martel, Merle and Raphaël. Due to strong interactions between the bubbles, it also relies decisively on the sharp properties of the minimal mass blow up solution (single bubble case) proved by the authors in arXiv:1602.03519.

70 pages