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Accepted
Lax pair of an integrable non-linear PDE
You could try using the Wahlquist-Estabrook prolongation structure technique, per H.D. Wahlquist and F.B. Estabrook, J. Math. Phys 16 (1975) 1-7 (covering the Korteweg-deVries equation), & F.B. ...
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Integrability - conditions of lax pairs
In addition to the answer by @issoloroap, the article Prolongation structures of nonlinear evolution equations by Allan Fordy (here is the Mathscinet link and here is its first page on Google books) ...
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1
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Accepted
Recovering the nonlinear Schrödinger equation from its Lax pair
Okay, here's the computation in detail. Really, this is a straightforward bracket computation, though it appears my comment earlier about a factor of 1/2 was in error (oops!) so no need to make a ...
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How to find a Lax Pair for the modified KdV equation
The point is that you should assume that $a,b,c$ are polynomials in $\lambda$ and then equate to zero the coefficients at various powers of $\lambda$.
Similar analysis can, I think, be found in many ...
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Lax pairs for linear PDEs
The term Lax pair is fairly often used in a broader sense as a pair of linear equations (not necessarily of the form you mention, i.e., $L\phi=\lambda\phi$ and $\dot{\phi}=B\phi$) whose compatibility ...
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