Modulation solutions for the Benjamin-One equation

In this work, modulation solutions for three initial value problems for the Benjamin-One equation are studied. The first problem studied is the disper sive resolution of a step initial condition. An explicit solution of Gurevi ch-Pitaevskii type is derived, which explains the dispersive resolution of the step in terms of modulations, The second problem is the dispersive reso lution of a breaking initial condition, while the third problem is the gene ration of a second phase as the result of the evolution of a modulated sing le phase wave. Again explicit modulation solutions for these problems are d erived. In the case of the third problem, the modulation solution explicitl y exhibits the formation of the new phase, in contrast to situation for the Kortweg-de Vries equation, for which the formation of a second phase has o nly been solved for phase loss in the dispersive analogue of shock merging. These explicit solutions are possible since the modulation theory of Dobro khotov and Krichever for the BO equation gives uncoupled modulation equatio ns for the different phases, which is not the case for the KdV equation. Th is 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 fo r the KdV equation for shock fomation and shock merging. (C) 1999 Elsevier Science B.V. All rights reserved.