 Solution

Use the Fourier convention ${\hat{q}}_0(k)={\int }_{\mathbb{R}}{e}^{-ikx}{q}_0(x)dx$ and sufficiently rapidly decaying smooth initial data. A scalar Lax pair for the linear equation isDifferentiate the first equation in time and use the second and its spatial derivative. The terms involving $m$ and $k$ cancel, leaving $q_t=i{q}_{xx}$, exactly the linear equation.

The spatially normalized solutions are $m^{+}={\int }_{-\infty }^x{e}^{ik(x-y)}q(y,t)dy$ for $\mathrm{Im}k>0$, and $m^{-}=-{\int }_x^{\infty }{e}^{ik(x-y)}q(y,t)dy$ for $\mathrm{Im}k<0$. Integration by parts gives $m^{\pm }=O(k^{-1})$. Their difference is $e^{ikx}\hat{q}(k,t)$; Fourier transformation of the evolution gives $\hat{q}(k,t)=e^{-i{k}^{2}t}{\hat{q}}_0(k)$. Set $M^{\pm }=\left(\begin{array}{cc}1& m^{\pm }\\ 0& 1\end{array}\right)$. ThenThe Cauchy integral solution is $m(k)=(2\pi i{)}^{-1}\int e^{isx-i{s}^{2}t}{\hat{q}}_0(s)/(s-k)ds$, and reconstruction $q=-i{\lim }_{k\to \infty }km(k)$ gives the usual inverse Fourier solution.

For the nonlinear deformation, put $D=\mathrm{diag}(1,-1)$ and $Q=\left(\begin{array}{cc}0& q\\ \vartheta & 0\end{array}\right)$. The matrix Lax pair isA direct multiplication gives the off-diagonal entries of $U_t-V_x+[U,V]$ as $q_t-i{q}_{xx}+2i{q}^2\vartheta$ and ${\vartheta }_t+i{\vartheta }_{xx}-2i{\vartheta }^{2}q$, with zero diagonal. Hence compatibility is precisely the two requested nonlinear equations.

Here is a normalized Riemann-Hilbert problem for this pair, in the case without discrete scattering poles. Let $a(k),b(k)$ be its two reflection data determined from the initial spatial scattering problem, and set $\theta =kx-k^{2}t$. Deform the triangular jump toThe data $a,b$ are time-independent in this convention; their spatial and temporal oscillations are displayed explicitly. Setting $b=0$ and $a={\hat{q}}_0$ recovers the linear problem. Thus the diagonal correction $-ab$ is necessary when the second off-diagonal jump is introduced; omitting it would destroy determinant one.

To verify the deformation rather than merely name it, note $J_x=(ik/2)[D,J]$ and $J_t=-(i{k}^2/2)[D,J]$. Therefore $(M_x-(ik/2)[D,M])M^{-1}$ and $(M_t+(i{k}^2/2)[D,M])M^{-1}$ have no jump. Analyticity and the expansion at infinity make them, by Liouville's theorem, respectively $Q$ and $-kQ+V_0$. The first expansion gives $Q=-(i/2)[D,M_1]$, soThe constant term of the second expression is $V_0=(M_1{)}_x$. In the first equation's next coefficient, its off-diagonal part equals $iD{Q}_x$ and its diagonal part equals $-iD{Q}^2$. Hence $V_0=iD(Q_x-Q^2)$, exactly the displayed nonlinear Lax pair. This proves that the reconstructed functions obey the required PDEs.

Finally let $C$ be the Cauchy integral operator on the real axis, $C_{-}$ its lower boundary value, and $\mu =M^{-}$. The jump relation and normalization giveThis is the requested linear singular integral equation for the inverse problem: it is linear in the unknown matrix $\mu$ once the scattering data are given. Reconstruction is $q=(2\pi {)}^{-1}\int [\mu (J-I){]}_{12}ds$ and $\vartheta =-(2\pi {)}^{-1}\int [\mu (J-I){]}_{21}ds$.

For completeness, generic simple discrete eigenvalues add the following residue data. At an upper-half-plane pole $k_j$, use $C_j=\left(\begin{array}{cc}0& c_j{e}^{i\theta (k_j)}\\ 0& 0\end{array}\right)$; at a lower-half-plane pole use $C_j=\left(\begin{array}{cc}0& 0\\ d_j{e}^{-i\theta (k_j)}& 0\end{array}\right)$, with the norming constants determined by the initial spatial scattering problem. Impose ${\mathrm{Res}}_{k_j}M={\lim }_{k\to k_j}M{C}_j$. The nilpotent matrices make this limit well-defined. If $A_j$ denotes the residue, the full linear inverse system isThe reconstructed field formulas acquire $-i{\sum }_j(A_j{)}_{12}$ and $i{\sum }_j(A_j{)}_{21}$ respectively. This includes the soliton terms when they occur; the preceding pure-jump formula is their pole-free special case. These standard scattering formulations assume no spectral singularity on the real contour and simple discrete eigenvalues; repeated poles require the corresponding higher-order pole conditions.