= Cauchy-kernel dressing for the KdV equation
{c}
{title2=$\varphi+iE_k\int_L\varphi(l)/(k+l)d\lambda(l)=E_k$}
Put $E_k=e^{i(kx+k^3t)}$, $I=\int_L\varphi(l)d\lambda(l)$ and $q=-I_x$. For the dressing operator $\mathcal Fh=h+iE_k\int_Lh(l)/(k+l)d\lambda(l)$, differentiating gives $\mathcal F(M\varphi)=0$ and $\mathcal F(N\varphi)=3ikM\varphi$, where $M=\partial_x^2-ik\partial_x+q$ and $N=\partial_t+\partial_x^3+3q\partial_x$. Uniqueness of the <linear integral equation> therefore supplies a <Lax pair>. Its <commutator> identity is $[N,M]=-3q_xM+(q_t+q_{xxx}+6qq_x)$, which gives the <KdV equation> when the eigenfunction is not identically zero. Differentiation under the integral and the uniqueness assumption are essential hypotheses.
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