Modulation solutions for the Benjamin–Ono equation

Abstract

In this work, modulation solutions for three initial value problems for the Benjamin–Ono equation are studied. The first problem studied is the dispersive resolution of a step initial condition. An explicit solution of Gurevich–Pitaevskii type is derived, which explains the dispersive resolution of the step in terms of modulations. The second problem is the dispersive resolution of a breaking initial condition, while the third problem is the generation of a second phase as the result of the evolution of a modulated single phase wave. Again explicit modulation solutions for these problems are derived. In the case of the third problem, the modulation solution explicitly exhibits the formation of the new phase, in contrast to situation for the Korteweg–de Vries equation, for which the formation of a second phase has only been solved for phase loss in the dispersive analogue of shock merging. These explicit solutions are possible since the modulation theory of Dobrokhotov and Krichever for the BO equation gives uncoupled modulation equations for the different phases, which is not the case for the KdV equation. This de-coupling means that the matching of known explicit solutions enables the full description of two-phase problems. This extends results which have recently been obtained from analytic solutions of the Whitham equations for the KdV equation for shock formation and shock merging.