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We show that the solution (in the sense of distribution) to the Cauchy problem with the periodic boundary condition associated with the modified Benjamin–Ono equation is unique in   $L^\infty _t(H^s(\mathbb{T} ))$   for   $s&gt;1/2$  . The proof is based on the analysis of a normal form equation obtained by infinitely many reduction steps using integration by parts in time after a suitable gauge transform.

Unconditional uniqueness for the periodic modified Benjamin–Ono equation by normal form approach