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Lax Pair

A Lax pair for a differential equation is a pair of linear operators L and A whose evolution satisfies the Lax equation

(dL)/(dt)=[A,L]=AL-LA,
(1)

where [A,L] is the commutator. This compatibility condition is equivalent to requiring the two equations

Lpsi=lambdapsi
(2)
(dpsi)/(dt)=Apsi.
(3)

to remain consistent as the system evolves. The Lax equation implies that the eigenvalues of L are conserved, providing the conserved quantities that help make the equation an integrable system.

It can be difficult to find a Lax pair for a given partial differential equation. Conversely, one may postulate L and A and determine the partial differential equation encoded by their compatibility condition (Infeld and Rowlands 2000).

See also

Commutator, Integrable System, Inverse Scattering Method, Partial Differential Equation

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References

Infeld, E. and Rowlands, G. "Integrable Equations in Two Space Dimensions as Treated by the Zakharov-Shabat Method." §7.10 in Nonlinear Waves, Solitons, and Chaos, 2nd ed. Cambridge, England: Cambridge University Press, pp. 192-199, 2000.

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Lax Pair

Cite this as:

Weisstein, Eric W. "Lax Pair." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LaxPair.html

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