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

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.