The Benjamin–Ono Equation in the Zero‐Dispersion Limit for Rational Initial Data: Generation of Dispersive Shock Waves

ABSTRACT The leading‐order asymptotic behavior of the solution of the Cauchy initial‐value problem for the Benjamin–Ono equation in is obtained explicitly for generic rational initial data . An explicit asymptotic wave profile is given, in terms of the branches of the multivalued solution of the inviscid Burgers equation with initial data , such that the solution of the Benjamin–Ono equation with dispersion parameter and initial data satisfies in the locally uniform sense as , provided a discriminant inequality holds implying that certain caustic curves in the ‐plane are avoided. In some cases, this convergence implies strong convergence. The asymptotic profile is consistent with the modulated multiphase wave solutions described by Dobrokhotov and Krichever.