What is the proof of the local well-posedness of the quadratic nonlinear Schrödinger equation $\mathrm{i} \,\partial_t u + \Delta u \pm \left|u\right| u = 0$ on the 1D torus in $H^s$ for $s > 1$ (a good reference would suffice)?

$H^s(\mathbb{T})$ is an algebra, but $\left|u\right| u$ is not of the form $u^2$, $\overline{u}u$ or $\overline{u}^2$ and so the LWP doest not immediately follow from the Banach contraction mapping principle.

The LWP should hold according to Tao's webpage (even for $s > 0$). However, the above problem is not covered by Theorem I in [Bo1993] (reference as on Tao's webpage) and in fact Remark (ii) after Proposition 5.73 states that uniqueness of the solutions is unclear.