Construction of infinite time bubble tower solutions for energy critical equivariant Schrödinger maps

We prove the existence of infinite time bubble tower solutions for energy critical $k$-equivariant Schrödinger maps ${\mathbb{R}}^2\to {\mathbb{S}}^2$ with $k\ge 3$. More precisely, for any integers $k\ge 3$ and $N\ge 1$, we construct a $k$-equivariant solution that is global in one time direction and decomposes asymptotically into a superposition of $N$ bubbles with scales ${\lambda }_j(t)\sim \mathrm{\mid }t{\mathrm{\mid }}^{-{\beta }_j}$, where ${\beta }_j=\frac{1}{2}({\left(\frac{k}{k-2}\right)}^{N-j}-1)$. Our approach is a backward construction combined with modulation analysis and high order energy estimates. We use Darboux coordinates adapted to the multi bubble profile to express the remainder in a complex scalar form. A key ingredient is the construction of a correction that cancels the leading effect of the nonlinear defect.