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In this paper, we revisit the infinite iteration scheme of normal formreductions, introduced by the first and second authors (with Z. Guo), inconstructing solutions to nonlinear dispersive PDEs. Our main goal is topresent a simplified approach to this method. More precisely, we study normalform reductions in an abstract form and reduce multilinear estimates ofarbitrarily high degrees to successive applications of basic trilinearestimates. As an application, we prove unconditional well-posedness ofcanonical nonlinear dispersive equations on the real line. In particular, weimplement this simplified approach to an infinite iteration of normal formreductions in the context of the cubic nonlinear Schr\"odinger equation (NLS)and the modified KdV equation (mKdV) on the real line and prove unconditionalwell-posedness in $H^s(\mathbb R)$ with (i) $s\geq \frac 16$ for the cubic NLSand (ii) $s > \frac 14$ for the mKdV. Our normal form approach also allows usto construct weak solutions to the cubic NLS in $H^s(\mathbb R)$, $0 \leq s <\frac 16$, and distributional solutions to the mKdV in $H^\frac{1}{4}(\mathbbR)$ (with some uniqueness statements).

Normal form approach to unconditional well-posedness of nonlinear dispersive PDEs on the real line