Full range of infinite point blow-up exponents for the critical generalized KdV equation

For the quintic, mass critical generalized Korteweg-de Vries equation, for any $\nu \in (\frac{1}{2},1)$, we prove the existence of solutions in the energy space that blow up in finite time $T>0$ with the blow-up rate $\mathrm{\parallel }{\mathrm{\partial }}_{x}u(t){\mathrm{\parallel }}_{L^2}\sim (T-t{)}^{-\nu }$ (infinite point blow-up). These solutions are constructed arbitrarily close to the family of solitons and correspond to the concentration of a soliton traveling at $+\mathrm{\infty }$ in space as $t\uparrow T$. This complements the previous results obtained in the work of Martel, Merle, Raphaël in 2015 on infinite point exotic blow-up, which were valid under the technical restriction $\nu >\frac{11}{13}$. The value $\nu =\frac{1}{2}$ corresponds to a critical case to be treated elsewhere. At the technical level, we implement a modification of the virial-energy functional, to allow all $\nu >\frac{1}{2}$ and simplify the proof of energy estimates.