= Finite-rank KdV dressing determinant
{title2=$q=2\partial_x^2\log\det(I+A)$}
Choose distinct $\kappa_j>0$ and weights $c_j>0$ at spectral points $i\kappa_j$. With $E_j=e^{-\kappa_jx+\kappa_j^3t}$ and $A_{ij}=c_jE_i/(\kappa_i+\kappa_j)$, solve $(I+A)\Phi=E$. The reconstructed <multisoliton solution> is $q=-\partial_x(c^T\Phi)$. Since $\Lambda A+A\Lambda=Ec^T$, cyclicity of the <trace> gives $c^T\Phi=-2\partial_x\log\det(I+A)$. The <Cauchy matrix> $1/(\kappa_i+\kappa_j)$ is positive definite, because it is the Gram matrix of $e^{-\kappa_i s}$ on $s>0$; diagonal similarity then proves $\det(I+A)>0$. For one spectral point this yields $q=(\kappa^2/2)\operatorname{sech}^2[\kappa(x-\kappa^2t-x_0)/2]$.
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