Pole dynamics and oscillations for the complex Burgers equation in the small-dispersion limit

D. Senouf, Russel Caflisch, N. Ercolani

Research output: Contribution to journalArticle

Abstract

A meromorphic solution to the Burgers equation with complex viscosity is analysed. The equation is linearized via the Cole-Hopf transform which allows for a careful study of the behaviour of the singularities of the solution. The asymptotic behaviour of the solution as the dispersion coefficient tends to zero is derived. For small dispersion, the time evolution of the poles is found by numerically solving a truncated infinite-dimensional Calogero-type dynamical system. The initial data are provided by high-order asymptotic approximations of the poles at the critical time ts for the dispersionless solution via the method of steepest descents. The solution is reconstructed using the pole expansion and the location of the poles. The oscillations observed via the singularities are compared to those obtained by a classical stationary phase analysis of the solution as the dispersion parameter ε → 0+. A uniform asymptotic expansion as ε → 0+ of the dispersive solution is derived in terms of the Pearcey integral in a neighbourhood of the caustic. A continuum limit of the pole expansion and the Calogero system is obtained, yielding a new integral representation of the solution to the inviscid Burgers equation.

Original languageEnglish (US)
Pages (from-to)1671-1702
Number of pages32
JournalNonlinearity
Volume9
Issue number6
DOIs
StatePublished - 1996

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Burger equation
Burgers Equation
Pole
Poles
poles
Oscillation
oscillations
expansion
Singularity
Uniform Asymptotic Expansion
Higher-order Asymptotics
Dispersion Parameter
Meromorphic Solution
Stationary Phase
Higher Order Approximation
Caustic
Steepest Descent
Continuum Limit
Asymptotic Approximation
Integral Representation

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

Cite this

Pole dynamics and oscillations for the complex Burgers equation in the small-dispersion limit. / Senouf, D.; Caflisch, Russel; Ercolani, N.

In: Nonlinearity, Vol. 9, No. 6, 1996, p. 1671-1702.

Research output: Contribution to journalArticle

Senouf, D. ; Caflisch, Russel ; Ercolani, N. / Pole dynamics and oscillations for the complex Burgers equation in the small-dispersion limit. In: Nonlinearity. 1996 ; Vol. 9, No. 6. pp. 1671-1702.
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