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Commuting flows and conservation laws for Lax equations

Published online by Cambridge University Press:  24 October 2008

George Wilson
George Wilson
Affiliation:
University College, London
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Extract

In recent years there has been great progress in the study of certain systems of non-linear partial differential equations, namely those that have a ‘Lax representation’

Here P and L are linear differential operators in one variable x, whose coefficients are l × l matrices of functions of x and t. Thus L has the form

where each ui is a matrix of functions ui,αβ(x, t), 1 ≤ α, β ≤ l. The symbol Lt means that we differentiate each coefficient of L, and as usual [P, L] = PL − LP. The coefficients of P are supposed to be polynomials in the ui, αβ and their x-derivatives, so that (1·1) is equivalent to a system of non-linear ‘evolution equations’ for the variables ui, αβ. The simplest example is the Korteweg–de Vries (KdV) equation

which has a Lax representation with

(Here l = 1, and there is only one coefficient ui, αβ. ) The connexion between the KdV equation and this ‘Schrodinger operator’ L was discovered by Gardner, Greene, Kruskal and Miura(6), but it was P. Lax (13) who first pointed out explicitly what we call the Lax representation given by (1·1) and (1·2). The notation (P, L), due to Gel'fand and Dikii(7), reflects this fact.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 86 , Issue 1 , July 1979 , pp. 131 - 143
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

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