A Justification of the KdV Approximation to First Order in the Case of N-Soliton Water Waves in a Canal.

Abstract

We consider the Euler equations for a perfect fluid in a flat-bottomed canal in the time-dependent case. A formal expansion procedure for small ampltitude, long waves analogous to that of Friedrichs and Hyers for solitary waves is developed and leads to the Korteweg-de Vries equation (KdV for short) for the lowest order term. The higher order terms in the expansion satisfy the inhomogenous version of the linearized KdV equation. Of particular interest to us are those solutions of the KdV equation called N-solitons, which asymptotically separate into N travelling waves with distinct speeds. Using certain facts about the linearized KdV equation and some properties of the N-solitons, we prove that the next term in this expansion can be uniquely specified by certain asymptotic conditions and a symmetry requirement. This solution behaves like an N-solition; asymptotically, it separates into N travelling waves with the same speeds and phases as those of the leading term. (Author)

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Document Details

Document Type
Technical Report
Publication Date
Apr 01, 1981
Accession Number
ADA100611

Entities

People

  • Robert L. Sachs

Organizations

  • University of Wisconsin–Madison

Tags

Communities of Interest

  • Air Platforms
  • Energy and Power Technologies

DTIC Thesaurus Topics

  • Amplitude
  • Boundaries
  • Cauchy Problem
  • Complex Variables
  • Computational Science
  • Equations
  • Euler Equations
  • Flow
  • Fluid Dynamics
  • Fluid Flow
  • Froude Number
  • Long Wavelengths
  • Mathematics
  • Rational Functions
  • Solitons
  • Two Dimensional
  • Water Waves

Fields of Study

  • Mathematics

Readers

  • Atmospheric Science / Meteorology, specifically Wind Wave Turbulence.
  • Finite Element Method (FEM) for solving Partial Differential Equations (PDEs)
  • Linear Algebra