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Dispersive shock wave interactions and asymptotics.

Mark J Ablowitz1, Douglas E Baldwin

  • 1Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309-0526, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 19, 2013
PubMed
Summary
This summary is machine-generated.

Dispersive shock waves (DSWs) in Korteweg-de Vries (KdV) systems evolve into a single-phase DSW over time. This phenomenon, observed for steplike data, represents the largest possible DSW determined by boundary conditions.

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Area of Science:

  • Physics
  • Nonlinear Dynamics
  • Fluid Dynamics

Background:

  • Dispersive shock waves (DSWs) are crucial in systems with weak dispersion and nonlinearity.
  • The Korteweg-de Vries (KdV) equation models such systems, particularly those with weak, quadratic nonlinearity.

Purpose of the Study:

  • To determine the long-time-asymptotic solution of the KdV equation for general steplike data.
  • To characterize the emergent structure of dispersive shock waves in nonlinear dispersive systems.

Main Methods:

  • Utilizing the inverse scattering transform to analyze the KdV equation.
  • Employing matched-asymptotic expansions to derive the long-time behavior.

Main Results:

  • The asymptotic solution for general steplike data is a single-phase DSW.
  • This emergent DSW is the largest possible, dictated by the boundary data.
  • Interacting multiphase dynamics from intermediate times merge into a single-phase DSW at large times.

Conclusions:

  • The long-time behavior of the KdV equation under steplike initial conditions is a single-phase DSW.
  • This finding unifies the understanding of complex dynamics in nonlinear dispersive systems.
  • The study highlights the universal tendency towards simpler, large-scale structures in such systems.