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 $L^2\left(R\right)$ is obtained explicitly for generic rational initial data $u_0$. An explicit asymptotic wave profile $u^{\mathrm{ZD}}(t,x;\epsilon )$ is given, in terms of the branches of the multivalued solution of the inviscid Burgers equation with initial data $u_0$, such that the solution $u(t,x;\epsilon )$ of the Benjamin–Ono equation with dispersion parameter $\epsilon >0$ and initial data $u_0$ satisfies $u(t,x;\epsilon )-u^{\mathrm{ZD}}(t,x;\epsilon )\to 0$ in the locally uniform sense as $\epsilon \to 0$, provided a discriminant inequality holds implying that certain caustic curves in the $(t,x)$-plane are avoided. In some cases, this convergence implies strong $L^2\left(R\right)$ convergence. The asymptotic profile $u^{\mathrm{ZD}}(t,x;\epsilon )$ is consistent with the modulated multiphase wave solutions described by Dobrokhotov and Krichever.