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Complex Singularities and the Riemann Surface for the Burgers Equation

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Nonlinear Physics

Part of the book series: Research Reports in Physics ((RESREPORTS))

Abstract

The analytic structure of the solution of the Burgers equations is analysed: the viscous solution has an infinite number of complex poles. When the viscosity tends to zero, these poles condense, producing the inviscid singularities. A Riemann surface is attached to those non polar singularities. As a consequence, a shock appears to be the permutation of two Riemann sheets. This phenomenon can also be understood as a phase transition in a Curie-Weiss model.

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References

  1. D. Bessis and J.D. Fourier, J.Phys.Let. 45, L833–841 (1984)

    Article  Google Scholar 

  2. C.M. Newman, private communications

    Google Scholar 

  3. J.A. Burgers, A.K. Verhaud-Kon, Nederl. Weterschappen Afd. Natuurkunde, Eerste Sertie, Vol. 17 (1939) p. 1–5

    Google Scholar 

  4. J.M. Burgers, “The Non-linear Diffusion equation” D. Reidel Publ. (1974)

    Google Scholar 

  5. C.M. Newman, Schock waves and linear field bound, Talk given at Rutgers Statistical Mechanic Meeting, Dec. 1981.

    Google Scholar 

  6. J.D. Fournier and U. Frisch, J.Mec.Th.Appl. 2 (1983) n°5, 689

    MathSciNet  Google Scholar 

  7. D.V. Choodnovsky and G.V. Choodnovsky, “Pole expansions of Non-linear Partial Differential equations” Nuovo Cimento 40B, n°2, p. 339–353 (1977)

    Article  MathSciNet  Google Scholar 

  8. J. Weiss, M. Tabor, G. Carnevale, “The Painlevé Property for Partial Differential Equations”, J.Math.Phys. 24, (3) p. 522–526 (1983)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. E. Hopf, Comm.Pure Appl.Mech. 3, 201 (1950).

    Article  MATH  MathSciNet  Google Scholar 

  10. J.D. Cole, Quart. Appl.Math. 9, 225 (1951)

    MATH  MathSciNet  Google Scholar 

  11. G. Polya, Uber trigonometische Integrale mit nur reellen Nullstelleir, Z. Reine Angew. Math. 158, 6–18, 1927

    Article  MATH  Google Scholar 

  12. M.J. Ablowitz and H. Segur, Solitons and the inverse scattering transforms, Siam 1981

    Google Scholar 

  13. T.D. Lee and C.N. Yang, Phys.Rev. 87, 410 (1952).

    Article  MATH  ADS  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Service de Physique Théorique de Saclay, Laboratoire de l’Institut de Recherche Fondamentale du Commissariat à l’Energie Atomique, F-91191, Gif-sur-Yvette Cedex, France

    D. Bessis

  2. Observatoire, Mont-Gros, BP 139, F-06003, Nice Cedex, France

    J. D. Fournier

Authors
  1. D. Bessis
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  2. J. D. Fournier
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Editor information

Editors and Affiliations

  1. University of Science and Technology of China, 230026, Hefei, Anhui, People’s Rep. of China

    Chaohao Gu  & Yishen Li  & 

  2. Computing Center of Academia Sinica, 100080, Beijing, People’s Rep. of China

    Guizhang Tu

  3. Department of Mathematics, University of Science and Technology of China, 230026, Hefei, Anhui, People’s Rep. of China

    Yunbo Zeng

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© 1990 Springer-Verlag Berlin, Heidelberg

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Bessis, D., Fournier, J.D. (1990). Complex Singularities and the Riemann Surface for the Burgers Equation. In: Gu, C., Li, Y., Tu, G., Zeng, Y. (eds) Nonlinear Physics. Research Reports in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-84148-4_27

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  • DOI: https://doi.org/10.1007/978-3-642-84148-4_27

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-52389-5

  • Online ISBN: 978-3-642-84148-4

  • eBook Packages: Springer Book Archive

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