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For real $q$, write $\bar{f}(k)=\overline{f(\bar{k})}$. The reduction symmetry and unit determinants give the scattering data parametrizationsTo see the symmetry, conjugate the coefficient matrices by $C=\left(\begin{array}{cc}0& 1\\ \lambda & 0\end{array}\right)$ after taking complex conjugates at $\bar{k}$; $C{\sigma }_3{C}^{-1}=-{\sigma }_3$ and the real-$q$ reduction restore the original Lax pair. DefineIf ${\mu }_j^{(l)}$ denotes column $l$, bounded columns produce the following normalized matrices:Each divisor is exactly the determinant of the unscaled column pair, so $\det M_j=1$. The eigenfunctions and divisors tend to $I$ and one, respectively, giving $M\to I$ at infinity. The contour for the sectorial half-line mKdV Riemann-Hilbert reconstruction isIt is six rays, not just the real axis or a four-quadrant contour. For the displayed meromorphic construction assume the divisors have no zeros on the contour; contour zeros require an indented or otherwise regularized formulation.

Here are explicit jump data without ambiguity about which side is used. DefineThe connection formulas give $M_j={\mu }_2{e}^{i\theta {\hat{\sigma }}_3}{C}_j$. Orient each ray outwards and let plus mean its left side. If that ray separates domains $D_i$ on the minus side and $D_j$ on the plus side, its jump isThis specifies every jump using only the spectral connection matrices; each jump has determinant one. Boundedness and consistency at the junction are inherited from the original eigenfunctions when the spectral data satisfy their global constraint.

If $a,\bar{a},d_{+}$ or $d_{-}$ has zeros in its corresponding domain, their simple zeros generate poles and require residue conditions. For example, at a simple zero $k_0$ of $a$ in $D_3$,At a simple zero of $d_{+}$ in $D_1$, the analogous coefficient for the first column is $\lambda \bar{b}{e}^{-2i\theta }/(A{d}_{+}^{\prime })$ times the second column; in $D_4$ it is $b{e}^{2i\theta }/(\bar{A}{d}_{-}^{\prime })$ for the second column. The $\bar{a}$ residue in $D_2$ is the conjugate first-column relation. These formulas assume the other displayed denominators are nonzero there; multiple or coincident zeros need their corresponding higher-order conditions. In a sufficiently small-data zero-free regime, the normalized Riemann-Hilbert problem has only the jumps. Finally the order-one term in ${\mu }_x-ik[{\sigma }_3,\mu ]=Q\mu$ givesThe column normalizations do not change this off-diagonal leading coefficient.

Eliminate ${\mu }_2$ from the two connection formulas:At $(0,T)$, where ${\mu }_1=I$ and $\theta =-4k^{3}T$, this is the half-line mKdV spectral global relationTaking its upper-right entry gives a particularly useful scalar form,The second column of ${\mu }_3$ shows that $c$ is analytic in the lower half-plane and has the usual $O(k^{-1})$ normalization. The opposite off-diagonal equation follows from the real-$q$ reduction. The exponential sign here follows from conjugating a $12$ entry, which multiplies it by $e^{8i{k}^{3}T}$; reversing it would give the wrong linear global relation and boundary count. This relation constrains the initial and boundary scattering data; they cannot be chosen independently.

Scattering data describe connections between normalized solutions of an auxiliary spectral equation, together with any discrete spectral points and their norming data. In an inverse scattering transform, the nonlinear potential is reconstructed from these data; boundary spectral connection matrices must additionally obey a global relation.

Use the bounded columns supplied by Volterra analyticity sectors for reverse-dispersion mKdV and divide the necessary columns by their determinants. This gives a normalized meromorphic matrix on the six-ray contour $\arg k=m\pi /3$. The constant scattering data connection matrices generate its jumps by conjugation with $e^{i(kx-4k^{3}t){\sigma }_3}$. Zeros of column determinants require residue conditions; in a sufficiently small-data, zero-free regime only jumps are needed. The off-diagonal expansion coefficient reconstructs $q=-2i{\lim }_{k\to \infty }k{M}_{12}$ for this sign convention.