Linearizable Robin scattering for defocusing NLS ($h(k)=(2k-ic)/(2k+ic)$)

For the Robin boundary condition $q_x(0,t)=cq(0,t)$ with real $c$, the time Lax pair has a diagonal-conjugation symmetry under $k\mapsto -k$. It implies $A(-k)=A(k)$ and $B(-k)=-h(k)B(k)$. Together with the half-line NLS global relation, this eliminates unknown boundary scattering from an equivalent Riemann-Hilbert problem determined by the initial data and $c$.

Here $\Gamma =B^{♯}/(ad)$ and $\Delta =\mathcal{D}/(2k+ic)$. The difference follows algebraically from the half-line NLS global relation and the Robin symmetry. Multiplication by the spatial-temporal jump factor gives $e^{2ikx-4i{k}^2(T-t)}$, decaying in the second quadrant for $t<T$. A triangular analytic change of the Riemann-Hilbert problem removes this finite-horizon defect while retaining the reconstruction and transferring all genuine pole information to the known determinant. No unproved large-time decay of the boundary values is needed.

The effective boundary coefficient is $\tilde{\Gamma }=-(2k-ic)b^{♯}(-k)/(a\mathcal{D})$. All its ingredients come from initial scattering and the Robin boundary condition. Genuine zeros produce pole data; common zeros in numerator and denominator can instead be removable. It is incorrect to assert absence of boundary poles for every real Robin parameter.