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We consider the zero-dispersion limit for the Benjamin-Ono equation on thetorus. We prove that when the initial data is a single well, thezero-dispersion limit exists in the weak sense and is uniform on every compacttime interval. Moreover, the limit is equal to the signed sum of branches forthe multivalued solution of the inviscid Burgers equation obtained by themethod of characteristics. This result is similar to the one obtained by Millerand Xu for the Benjamin-Ono equation on the real line for decaying and positiveinitial data. We also establish some precise asymptotics of the spectral datawith the cosine initial data, justifying our approximation method, which isanalogous to the work of Miller and Wetzel concerning a family of rationalpotentials for the Benjamin-Ono equation on the real line.

Zero-dispersion limit for the Benjamin-Ono equation on the torus with single well initial data