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.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. Bessis and J.D. Fourier, J.Phys.Let. 45, L833–841 (1984)
C.M. Newman, private communications
J.A. Burgers, A.K. Verhaud-Kon, Nederl. Weterschappen Afd. Natuurkunde, Eerste Sertie, Vol. 17 (1939) p. 1–5
J.M. Burgers, “The Non-linear Diffusion equation” D. Reidel Publ. (1974)
C.M. Newman, Schock waves and linear field bound, Talk given at Rutgers Statistical Mechanic Meeting, Dec. 1981.
J.D. Fournier and U. Frisch, J.Mec.Th.Appl. 2 (1983) n°5, 689
D.V. Choodnovsky and G.V. Choodnovsky, “Pole expansions of Non-linear Partial Differential equations” Nuovo Cimento 40B, n°2, p. 339–353 (1977)
J. Weiss, M. Tabor, G. Carnevale, “The Painlevé Property for Partial Differential Equations”, J.Math.Phys. 24, (3) p. 522–526 (1983)
E. Hopf, Comm.Pure Appl.Mech. 3, 201 (1950).
J.D. Cole, Quart. Appl.Math. 9, 225 (1951)
G. Polya, Uber trigonometische Integrale mit nur reellen Nullstelleir, Z. Reine Angew. Math. 158, 6–18, 1927
M.J. Ablowitz and H. Segur, Solitons and the inverse scattering transforms, Siam 1981
T.D. Lee and C.N. Yang, Phys.Rev. 87, 410 (1952).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer-Verlag Berlin, Heidelberg
About this paper
Cite this paper
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
Download citation
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