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Theoretical and Mathematical Physics

, Volume 153, Issue 1, pp 1373–1387 | Cite as

Model equation of the theory of solitons

  • V. É. Adler
  • A. B. Shabat
Article

Abstract

We consider the hierarchy of integrable (1+2)-dimensional equations related to the Lie algebra of vector fields on the line. We construct solutions in quadratures that contain n arbitrary functions of a single argument. A simple equation for the generating function of the hierarchy, which determines the dynamics in negative times and finds applications to second-order spectral problems, is of main interest. Considering its polynomial solutions under the condition that the corresponding potential is regular allows developing a rather general theory of integrable (1+1)-dimensional equations.

Keywords

hierarchy of commuting vector fields Riemann invariant Dubrovin equations 

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Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Landau Institute for Theoretical PhysicsRASChernogolovka, Moscow OblastRussia

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