Inverse scattering transform

The inverse scattering transform evolves the scattering data of an auxiliary linear operator and reconstructs the potential from the evolved data.

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.

Put $E_k=e^{i(kx+k^{3}t)}$, $I={\int }_L\phi (l)d\lambda (l)$ and $q=-I_x$. For the dressing operator $\mathcal{F}h=h+i{E}_k{\int }_{L}h(l)/(k+l)d\lambda (l)$, differentiating gives $\mathcal{F}(M\phi )=0$ and $\mathcal{F}(N\phi )=3ikM\phi$, where $M={\partial }_x^2-ik{\partial }_x+q$ and $N={\partial }_t+{\partial }_x^3+3q{\partial }_x$. Uniqueness of the linear integral equation therefore supplies a Lax pair. Its commutator identity is $[N,M]=-3q_{x}M+(q_t+q_{xxx}+6q{q}_x)$, which gives the KdV equation when the eigenfunction is not identically zero. Differentiation under the integral and the uniqueness assumption are essential hypotheses.

Choose distinct ${\kappa }_j>0$ and weights $c_j>0$ at spectral points $i{\kappa }_j$. With $E_j=e^{-{\kappa }_{j}x+{\kappa }_j^{3}t}$ and $A_{ij}=c_j{E}_i/({\kappa }_i+{\kappa }_j)$, solve $(I+A)\Phi =E$. The reconstructed multisoliton solution is $q=-{\partial }_x(c^T\Phi )$. Since $\Lambda A+A\Lambda =E{c}^T$, cyclicity of the trace gives $c^T\Phi =-2{\partial }_x\log \det (I+A)$. The Cauchy matrix $1/({\kappa }_i+{\kappa }_j)$ is positive definite, because it is the Gram matrix of $e^{-{\kappa }_{i}s}$ on $s>0$; diagonal similarity then proves $\det (I+A)>0$. For one spectral point this yields $q=({\kappa }^2/2){\mathrm{sech}}^2[\kappa (x-{\kappa }^{2}t-x_0)/2]$.

Discrete scattering data consist of the discrete eigenvalues and norming constants associated with bound states of the auxiliary spectral problem. Their simple time evolution reconstructs solitons and breathers through the inverse problem.

For one-dimensional inverse scattering, the Gelfand-Levitan-Marchenko equationrecovers the transformation kernel $K$ from scattering data encoded by $F$. In the convention used for the Korteweg-De Vries equation above, the potential is

If $F(s)=c{e}^{-\chi s}$ with $\chi >0$, the Gelfand-Levitan-Marchenko equation has a separable kernel. Writing $K(x,y)=-B(x)e^{-\chi y}$ reduces the integral equation to one algebraic equation for $B(x)$ and reconstructs a single soliton.