Growth processes related to the dispersionless Lax equations

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8 Citations (Scopus)


This paper is a short review of the connection between certain types of growth processes and the integrable systems theory, written from the viewpoint of the latter. Starting from the dispersionless Lax equations for the 2D Toda hierarchy, we interpret them as evolution equations for conformal maps in the plane. This provides a unified approach to evolution of smooth domains (such as Laplacian growth) and growth of slits. We show that the Löwner differential equation for a parametric family of conformal maps of slit domains arises as a consistency condition for reductions of the dispersionless Toda hierarchy. It is also demonstrated how the both types of growth processes can be simulated by the large N limit of the Dyson gas picture for the model of normal random matrices.

Original languageEnglish
Pages (from-to)101-108
Number of pages8
JournalPhysica D: Nonlinear Phenomena
Issue number1-2 SPEC. ISS.
Publication statusPublished - Nov 2007
Externally publishedYes


  • Conformal maps
  • Dispersionless limit
  • Laplacian growth
  • Lax equations
  • Loewner equation


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