Riemann-Hilbert problem ($F_{+}-F_{-}=J$)

Find functions analytic on the two sides of a contour whose boundary values obey a prescribed jump, with suitable normalization at infinity and other distinguished points. An additive jump $J$ on a counterclockwise closed contour is solved by the Cauchy integral formula $F(k)=(2\pi i{)}^{-1}\int J(\zeta )/(\zeta -k)d\zeta$, subject to the normalization and compatibility conditions. The Sokhotski–Plemelj theorem verifies the jump.

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.

Combine columns of the three normalized Lax pair eigenfunctions in their common Volterra analyticity sectors for a half-line NLS Lax pair, dividing by the relevant determinants $a,d,d^{♯},a^{♯}$. The resulting matrix is normalized to the identity at infinity and has jumps on the real and imaginary axes. Connection matrices determine each jump explicitly. Zeros of those determinants require residue conditions or local contour deformations.

Suppose a scalar canonical factor has exterior growth $X_{-}(z)\sim z^q$ with index $-q<0$, and the original exterior unknown must decay as $O(z^{-1})$. Dividing by the canonical factor gives an additive jump $D_{+}-D_{-}=g$ with exterior decay $O(z^{-q-1})$. The exterior Cauchy integral expands in moments $-{\int }_L{t}^{j}g(t)dt/(2\pi i{z}^{j+1})$. Its first $q$ coefficients must therefore vanish, yielding the displayed solvability conditions. Under this decay there is no nonzero entire polynomial addition. A nonlocal moment appearing in $g$ must additionally agree with the moment of the reconstructed original jump.

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 $e^{-i(kx+2k^{2}t){\sigma }_3}$. 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 $i{Q}_t-Q_{xx}{\sigma }_3+2Q^3{\sigma }_3=0$. 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.

A rational jump with no zeros or poles on the contour can be split into factors analytic and nonvanishing on the two sides. The exterior factor may carry a power of $z$ recording the winding number. For a circle enclosing $0,\pm 1$, the jump $G=z/(z^2-1)$ has the canonical factors $X_{+}=1$, $X_{-}=z-z^{-1}$. The exterior growth of $X_{-}$ imposes compatibility conditions when a Cauchy transform is required to vanish at infinity.