Evolution of initial discontinuities in the Riemann problem for the Kaup-Boussinesq equation with positive dispersion

T. Congy 1 S. K. Ivanov 2, 3 A. M. Kamchatnov 4 N. Pavloff 1

Chaos, American Institute of Physics, 2017, 27 (8), pp.083107. 〈10.1063/1.4997052〉

We consider the space-time evolution of initial discontinuities of depth and flow velocity for an integrable version of the shallow water Boussinesq system introduced by Kaup. We focus on a specific version of this « Kaup-Boussinesq model » for which a flat water surface is modulationally stable, we speak below of « positive dispersion » model. This model also appears as an approximation to the equations governing the dynamics of polarisation waves in two-component Bose-Einstein condensates. We describe its periodic solutions and the corresponding Whitham modulation equations. The self-similar, one-phase wave structures are composed of different building blocks which are studied in detail. This makes it possible to establish a classification of all the possible wave configurations evolving from initial discontinuities. The analytic results are confirmed by numerical simulations.

  • 1. LPTMS – Laboratoire de Physique Théorique et Modèles Statistiques
  • 2. MIPT – Moscow Institute of Physics and Technology [Moscow]
  • 3. Institute of Spectroscopy
  • 4. Institute of Spectroscopy

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