Zero-Dispersion Limit for the Benjamin–Ono Equation on the Torus with Bell Shaped Initial Data

We consider the zero-dispersion limit for Benjamin–Ono equation on torus. prove that when initial data is bell shaped, exists in weak sense and uniform every compact time interval. Moreover, equal to signed sum of branches multivalued solution inviscid Burgers obtained by method characteristics. This result similar one Miller Xu real line decaying positive data. also establish some precise asymptotics spectral with $$u_0(x)=-\beta \cos (x)$$ , $$\beta >0$$ justifying our approximation method, which analogous work Wetzel concerning a family rational potentials line.