This article is cited in 20 papers

Model equation of the theory of solitons

V. E. Adler, A. B. Shabat

L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences

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.

Received: 29.01.2007

DOI: 10.4213/tmf6119

 English version: Theoretical and Mathematical Physics, 2007, 153:1, 1373–1387

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