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There is a literal inconsistency in the stated Riemann-Hilbert problem: a jump matrix $S(k)$ tending to zero is incompatible with ${\mu }_{\pm }\to I$. Along the real axis the conjugating exponentials are unitary, so the right side of the jump tends to zero while the left side tends to $I$. No normalized solution satisfies the printed decay condition. The intended condition is $S(k)\to I$, or replacement of $S$ in the jump by $I+S$ when $S$ denotes its decaying part.

For this corrected problem assume a regular, invertible, pole-free solution, differentiable in $x,t$, with the required expansion uniformly at infinity. These conditions matter: arbitrary jump data need not be solvable, and poles would contribute extra terms. Write $\sigma ={\sigma }_3$ and $\mu =I+{\mu }_1/k+{\mu }_2/k^2+\cdots$. Matching the nondecaying coefficients in the two differential expressions yieldsThe first expression makes $Q$ off-diagonal.

These matches really establish the differential equations, rather than just formal identities. Let $G=e^{-i(kx+2k^{2}t)\sigma }S(k)e^{i(kx+2k^{2}t)\sigma }$. It satisfies $G_x=-ik[\sigma ,G]$ and $G_t=-2i{k}^2[\sigma ,G]$. Differentiating ${\mu }_{+}={\mu }_{-}G$ consequently shows thathave matching boundary values. They therefore extend holomorphically across the contour. The chosen coefficients make both residuals $O(k^{-1})$ at infinity. Their absence of poles and Liouville theorem imply $R_x=R_t=0$, proving the required Lax pair under the corrected hypotheses. This is matrix NLS reconstruction from a normalized Riemann-Hilbert problem.