= Matrix NLS reconstruction from a normalized Riemann-Hilbert problem
{title2=$Q=i[\sigma_3,\mu_1],\qquad B=-2\partial_x\mu_1=-i(Q^2+Q_x)\sigma_3$}
Assume a regular invertible normalized <Riemann-Hilbert problem> solution $\mu=I+\mu_1/k+\mu_2/k^2+\cdots$ with jump tending to the identity and conjugated by $e^{-i(kx+2k^2t)\sigma_3}$. Residuals of its two differential equations have no jump and vanish at infinity, so <Liouville theorem> gives a <Lax pair>. Coefficient comparison gives the displayed reconstruction and $A=2Q$; compatibility yields $iQ_t-Q_{xx}\sigma_3+2Q^3\sigma_3=0$. A jump matrix tending to zero instead is inconsistent with the normalization. Recovering the initial jump from a prescribed initial potential still requires direct scattering.
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