A pair of linear operators and associated with a given partial differential equation which can be used to solve the equation. However, it turns out to be very difficult to find the and corresponding to a given equation, so it is actually simpler to postulate a given and and determine to which partial differential equation they correspond (Infeld and Rowlands 2000).

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