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Dispersive shock wave interactions and asymptotics

Mark J. Ablowitz and Douglas E. Baldwin*

  • Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309-0526, USA

  • *dsw@douglasbaldwin.com; http://www.douglasbaldwin.com
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Phys. Rev. E 87, 022906 – Published 13 February, 2013

DOI: https://doi.org/10.1103/PhysRevE.87.022906

Abstract

Dispersive shock waves (DSWs) are physically important phenomena that occur in systems dominated by weak dispersion and weak nonlinearity. The Korteweg–de Vries (KdV) equation is the universal model for systems with weak dispersion and weak, quadratic nonlinearity. Here we show that the long-time-asymptotic solution of the KdV equation for general, steplike data is a single-phase DSW; this DSW is the “largest” possible DSW based on the boundary data. We find this asymptotic solution using the inverse scattering transform and matched-asymptotic expansions. So while multistep data evolve to have multiphase dynamics at intermediate times, these interacting DSWs eventually merge to form a single-phase DSW at large time.

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Phys. Rev. E 87, 022906– Published 13 February, 2013
  • Received 9 April 2012
Article part of CHORUS

©2013 American Physical Society

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