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Whitham modulation theory for the two-dimensional Benjamin-Ono equation.

Mark Ablowitz1, Gino Biondini2,3, Qiao Wang2

  • 1Department of Applied Mathematics, State University of Colorado, Boulder, Colorado 80303, USA.

Physical Review. E
|January 20, 2018
PubMed
Summary
This summary is machine-generated.

Whitham modulation theory is applied to the 2D Benjamin-Ono equation, yielding a new system for wave modulations. This system aids in analyzing the transverse stability of traveling waves.

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Area of Science:

  • Fluid Dynamics and Nonlinear Waves
  • Mathematical Physics

Background:

  • The Benjamin-Ono (2DBO) equation is a fundamental model in nonlinear wave phenomena.
  • Understanding the stability and modulation of its traveling wave solutions is crucial.

Purpose of the Study:

  • To develop and analyze a Whitham modulation theory for the two-dimensional Benjamin-Ono (2DBO) equation.
  • To investigate the transverse stability of traveling wave solutions within the 2DBO framework.

Main Methods:

  • Derivation of a system of five quasilinear first-order partial differential equations.
  • Transformation of these equations into a singularity-free hydrodynamic-like system (2DBO-Whitham system).
  • Analysis of exact reductions and formulation of initial value problems for the derived system.

Main Results:

  • A novel 2DBO-Whitham system is established to describe wave modulations.
  • The system provides a framework for studying the transverse stability of 2DBO traveling waves.
  • Exact reductions and initial value problem formulations are presented.

Conclusions:

  • The developed 2DBO-Whitham system offers a powerful tool for analyzing nonlinear wave dynamics.
  • This approach facilitates a deeper understanding of the stability properties of traveling waves in the 2DBO equation.