KdV scattering data

For a rapidly decaying KdV potential, the scattering data consist of the continuous reflection coefficient, the negative discrete eigenvalues $-{\kappa }_n^2$, and a positive norming constant for each bound state.

For $u_t-6u{u}_x+u_{xxx}=0$, the discrete eigenvalues are fixed, whilewhen $c_n$ denotes the bound-state asymptotic amplitude. The squared amplitudes appearing in the Marchenko kernel consequently evolve by $e^{8{\kappa }_n^{3}t}$.

For a weak potential $u=ϵq$, the first Born approximation in the KdV Schrödinger convention is